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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. A topological vector space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
that is also a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
with the property that the vector space operations (vector addition and scalar multiplication) are also
continuous functions In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
and completeness. Some authors also require that the space is a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
(although this article does not). One of the most widely studied categories of TVSs are
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
s. This article focuses on TVSs that are not necessarily locally convex.
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s,
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s and
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
s are other well-known examples of TVSs. Many topological vector spaces are spaces of
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s, or
linear operators In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Wei ...
of sequences of functions. In this article, the
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
field of a topological vector space will be assumed to be either the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s \Complex or the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s \R, unless clearly stated otherwise.


Motivation


Normed spaces

Every
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
has a natural
topological structure In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
: the norm induces a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
and the metric induces a topology. This is a topological vector space because: #The vector addition map \cdot\, + \,\cdot\; : X \times X \to X defined by (x, y) \mapsto x + y is (jointly) continuous with respect to this topology. This follows directly from the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
obeyed by the norm. #The scalar multiplication map \cdot : \mathbb \times X \to X defined by (s, x) \mapsto s \cdot x, where \mathbb is the underlying scalar field of X, is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm. Thus all
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s and
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s are examples of topological vector spaces.


Non-normed spaces

There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s on an open domain, spaces of
infinitely differentiable function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if i ...
s, the
Schwartz space In mathematics, Schwartz space \mathcal is the function space of all Function (mathematics), functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. T ...
s, and spaces of
test function Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
s and the spaces of distributions on them. These are all examples of
Montel space In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector ...
s. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by
Kolmogorov's normability criterion In mathematics, Kolmogorov's normability criterion is a theorem that provides a necessary and sufficient condition for a topological vector space to be ; that is, for the existence of a norm on the space that generates the given topology. The norma ...
. A
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
is a topological vector space over each of its subfields.


Definition

A topological vector space (TVS) X is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
\mathbb (most often the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers with their standard topologies) that is endowed with a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
such that vector addition \cdot\, + \,\cdot\; : X \times X \to X and scalar multiplication \cdot : \mathbb \times X \to X are
continuous functions In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
(where the domains of these functions are endowed with product topologies). Such a topology is called a or a on X. Every topological vector space is also a commutative
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
under addition. Hausdorff assumption Many authors (for example,
Walter Rudin Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Walter Jacobs (1930–1968) * Gunther (wrestler), Austrian professional wrestler and trainer Walter Hahn (born 19 ...
), but not this page, require the topology on X to be T1; it then follows that the space is Hausdorff, and even Tychonoff. A topological vector space is said to be if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
. Category and morphisms The
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of topological vector spaces over a given topological field \mathbb is commonly denoted \mathrm_\mathbb or \mathrm_\mathbb. The
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
are the topological vector spaces over \mathbb and the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s are the continuous \mathbb-linear maps from one object to another. A (abbreviated ), also called a , is a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
u : X \to Y between topological vector spaces (TVSs) such that the induced map u : X \to \operatorname u is an
open mapping In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
when \operatorname u := u(X), which is the range or image of u, is given the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced by Y. A (abbreviated ), also called a , is an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a
topological embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...
. A (abbreviated ), also called a or an , is a bijective
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
. Equivalently, it is a
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
TVS embedding Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. A necessary condition for a vector topology A collection \mathcal of subsets of a vector space is called if for every N \in \mathcal, there exists some U \in \mathcal such that U + U \subseteq N. All of the above conditions are consequently a necessity for a topology to form a vector topology.


Defining topologies using neighborhoods of the origin

Since every vector topology is translation invariant (which means that for all x_0 \in X, the map X \to X defined by x \mapsto x_0 + x is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
), to define a vector topology it suffices to define a
neighborhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
(or subbasis) for it at the origin. In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.


Defining topologies using strings

Let X be a vector space and let U_ = \left(U_i\right)_^ be a sequence of subsets of X. Each set in the sequence U_ is called a of U_ and for every index i, U_i is called the i-th knot of U_. The set U_1 is called the beginning of U_. The sequence U_ is/is a: * if U_ + U_ \subseteq U_i for every index i. *
Balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
(resp. absorbing, closed,The topological properties of course also require that X be a TVS. convex, open,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, barrelled, absolutely convex/disked, etc.) if this is true of every U_i. * if U_ is summative, absorbing, and balanced. * or a in a TVS X if U_ is a string and each of its knots is a neighborhood of the origin in X. If Uis an absorbing
disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...
in a vector space X then the sequence defined by U_i := 2^ U forms a string beginning with U_1 = U. This is called the natural string of U Moreover, if a vector space X has countable dimension then every string contains an
absolutely convex In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull ...
string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued
subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
functions. These functions can then be used to prove many of the basic properties of topological vector spaces. A proof of the above theorem is given in the article on metrizable topological vector spaces. If U_ = \left(U_i\right)_ and V_ = \left(V_i\right)_ are two collections of subsets of a vector space X and if s is a scalar, then by definition: * V_ contains U_: \ U_ \subseteq V_ if and only if U_i \subseteq V_i for every index i. * Set of knots: \ \operatorname U_ := \left\. * Kernel: \ \ker U_ := \bigcap_ U_i. * Scalar multiple: \ s U_ := \left(s U_i\right)_. * Sum: \ U_ + V_ := \left(U_i + V_i\right)_. * Intersection: \ U_ \cap V_ := \left(U_i \cap V_i\right)_. If \mathbb is a collection sequences of subsets of X, then \mathbb is said to be directed (downwards) under inclusion or simply directed downward if \mathbb is not empty and for all U_, V_ \in \mathbb, there exists some W_ \in \mathbb such that W_ \subseteq U_ and W_ \subseteq V_ (said differently, if and only if \mathbb is a
prefilter In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then ...
with respect to the containment \,\subseteq\, defined above). Notation: Let \operatorname \mathbb := \bigcup_ \operatorname U_ be the set of all knots of all strings in \mathbb. Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. If \mathbb is the set of all topological strings in a TVS (X, \tau) then \tau_ = \tau. A Hausdorff TVS is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
if and only if its topology can be induced by a single topological string.


Topological structure

A vector space is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by -1). Hence, every topological vector space is an abelian
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
. Every TVS is
completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
but a TVS need not be
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
. Let X be a topological vector space. Given a subspace M \subseteq X, the quotient space X / M with the usual
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
is a Hausdorff topological vector space if and only if M is closed.In particular, X is Hausdorff if and only if the set \ is closed (that is, X is a T1 space). This permits the following construction: given a topological vector space X (that is probably not Hausdorff), form the quotient space X / M where M is the closure of \. X / M is then a Hausdorff topological vector space that can be studied instead of X.


Invariance of vector topologies

One of the most used properties of vector topologies is that every vector topology is : :for all x_0 \in X, the map X \to X defined by x \mapsto x_0 + x is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
, but if x_0 \neq 0 then it is not linear and so not a TVS-isomorphism. Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if s \neq 0 then the linear map X \to X defined by x \mapsto s x is a homeomorphism. Using s = -1 produces the negation map X \to X defined by x \mapsto - x, which is consequently a linear homeomorphism and thus a TVS-isomorphism. If x \in X and any subset S \subseteq X, then \operatorname_X (x + S) = x + \operatorname_X S and moreover, if 0 \in S then x + S is a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
(resp. open neighborhood, closed neighborhood) of x in X if and only if the same is true of S at the origin.


Local notions

A subset E of a vector space X is said to be * absorbing (in X): if for every x \in X, there exists a real r > 0 such that c x \in E for any scalar c satisfying , c, \leq r. *
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
or circled: if t E \subseteq E for every scalar , t, \leq 1. *
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
: if t E + (1 - t) E \subseteq E for every real 0 \leq t \leq 1. * a
disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...
or
absolutely convex In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull ...
: if E is convex and balanced. *
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
: if - E \subseteq E, or equivalently, if - E = E. Every neighborhood of the origin is an
absorbing set In functional analysis and related areas of mathematics an absorbing set in a vector space is a Set (mathematics), set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are ...
and contains an open
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
neighborhood of 0 so every topological vector space has a local base of absorbing and
balanced set In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \le ...
s. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of 0; if the space is
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin. Bounded subsets A subset E of a topological vector space X is
bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
if for every neighborhood V of the origin, then E \subseteq t V when t is sufficiently large. The definition of boundedness can be weakened a bit; E is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set. Also, E is bounded if and only if for every balanced neighborhood V of the origin, there exists t such that E \subseteq t V. Moreover, when X is locally convex, the boundedness can be characterized by
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
s: the subset E is bounded if and only if every continuous seminorm p is bounded on E. Every
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
set is bounded. If M is a vector subspace of a TVS X, then a subset of M is bounded in M if and only if it is bounded in X.


Metrizability

A TVS is pseudometrizable if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an ''F''-seminorm. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable. More strongly: a topological vector space is said to be
normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin. Let \mathbb be a non-
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
topological field, for example the real or complex numbers. A Hausdorff topological vector space over \mathbb is locally compact if and only if it is
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
, that is, isomorphic to \mathbb^n for some natural number n.


Completeness and uniform structure

The canonical uniformity on a TVS (X, \tau) is the unique translation-invariant
uniformity Uniformity may refer to: * Distribution uniformity, a measure of how uniformly water is applied to the area being watered * Religious uniformity, the promotion of one state religion, denomination, or philosophy to the exclusion of all other relig ...
that induces the topology \tau on X. Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s. This allows one to about related notions such as completeness,
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
, Cauchy nets, and
uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is Tychonoff. A subspace of a TVS is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
if and only if it is complete and
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
(for Hausdorff TVSs, a set being totally bounded is equivalent to it being precompact). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are
relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since ...
). With respect to this uniformity, a
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
(or
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
) x_ = \left(x_i\right)_ is Cauchy if and only if for every neighborhood V of 0, there exists some index n such that x_i - x_j \in V whenever i \geq n and j \geq n. Every Cauchy sequence is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called
sequentially complete In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in . is called sequentially complete if i ...
; in general, it may not be complete (in the sense that all Cauchy filters converge). The vector space operation of addition is uniformly continuous and an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...
. Scalar multiplication is Cauchy continuous but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of a
complete topological vector space In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
. * Every TVS has a completion and every Hausdorff TVS has a Hausdorff completion. Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions. * A compact subset of a TVS (not necessarily Hausdorff) is complete. A complete subset of a Hausdorff TVS is closed. * If C is a complete subset of a TVS then any subset of C that is closed in C is complete. * A Cauchy sequence in a Hausdorff TVS X is not necessarily
relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since ...
(that is, its closure in X is not necessarily compact). * If a Cauchy filter in a TVS has an
accumulation point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
x then it converges to x. * If a series \sum_^ x_i convergesA series \sum_^ x_i is said to converge in a TVS X if the sequence of partial sums converges. in a TVS X then x_ \to 0 in X.


Examples


Finest and coarsest vector topology

Let X be a real or complex vector space. Trivial topology The
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
or indiscrete topology \ is always a TVS topology on any vector space X and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on X always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
)
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
pseudometrizable seminormable
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
topological vector space. It is Hausdorff if and only if \dim X = 0. Finest vector topology There exists a TVS topology \tau_f on X, called the on X, that is finer than every other TVS-topology on X (that is, any TVS-topology on X is necessarily a subset of \tau_f). Every linear map from \left(X, \tau_f\right) into another TVS is necessarily continuous. If X has an uncountable
Hamel basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as component ...
then \tau_f is
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
and
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
.


Cartesian products

A
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of a family of topological vector spaces, when endowed with the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
, is a topological vector space. Consider for instance the set X of all functions f: \R \to \R where \R carries its usual
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
. This set X is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
\R^\R,, which carries the natural
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. With this product topology, X := \R^ becomes a topological vector space whose topology is called The reason for this name is the following: if \left(f_n\right)_^ is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
(or more generally, a
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
) of elements in X and if f \in X then f_n converges to f in X if and only if for every real number x, f_n(x) converges to f(x) in \R. This TVS is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
, Hausdorff, and
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
but not
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
and consequently not
normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form \R f := \ with f \neq 0).


Finite-dimensional spaces

By
F. Riesz's theorem F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences ...
, a Hausdorff topological vector space is finite-dimensional if and only if it is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
, which happens if and only if it has a compact
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the origin. Let \mathbb denote \R or \Complex and endow \mathbb with its usual Hausdorff normed
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
. Let X be a vector space over \mathbb of finite dimension n := \dim X and so that X is vector space isomorphic to \mathbb^n (explicitly, this means that there exists a
linear isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
between the vector spaces X and \mathbb^n). This finite-dimensional vector space X always has a unique vector topology, which makes it TVS-isomorphic to \mathbb^n, where \mathbb^n is endowed with the usual Euclidean topology (which is the same as the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
). This Hausdorff vector topology is also the (unique) finest vector topology on X. X has a unique vector topology if and only if \dim X = 0. If \dim X \neq 0 then although X does not have a unique vector topology, it does have a unique vector topology. * If \dim X = 0 then X = \ has exactly one vector topology: the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
, which in this case (and in this case) is Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension 0. * If \dim X = 1 then X has two vector topologies: the usual
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
and the (non-Hausdorff) trivial topology. ** Since the field \mathbb is itself a 1-dimensional topological vector space over \mathbb and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an
absorbing set In functional analysis and related areas of mathematics an absorbing set in a vector space is a Set (mathematics), set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are ...
and has consequences that reverberate throughout
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. * If \dim X = n \geq 2 then X has distinct vector topologies: ** Some of these topologies are now described: Every linear functional f on X, which is vector space isomorphic to \mathbb^n, induces a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
, f, : X \to \R defined by , f, (x) = , f(x), where \ker f = \ker , f, . Every seminorm induces a ( pseudometrizable
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
) vector topology on X and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on X that are induced by linear functionals with distinct kernel will induces distinct vector topologies on X. ** However, while there are infinitely many vector topologies on X when \dim X \geq 2, there are, only 1 + \dim X vector topologies on X. For instance, if n := \dim X = 2 then the vector topologies on X consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on X are all TVS-isomorphic to one another.


Non-vector topologies

Discrete and cofinite topologies If X is a non-trivial vector space (that is, of non-zero dimension) then the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
on X (which is always
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
) is a TVS topology because despite making addition and negation continuous (which makes it into a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
under addition), it fails to make scalar multiplication continuous. The
cofinite topology In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
on X (where a subset is open if and only if its complement is finite) is also a TVS topology on X.


Linear maps

A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator f is continuous if f(X) is bounded (as defined below) for some neighborhood X of the origin. A
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
on a topological vector space X is either dense or closed. A
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
f on a topological vector space X has either dense or closed kernel. Moreover, f is continuous if and only if its
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
.


Types

Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the
closed graph theorem In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. Graphs and map ...
, the open mapping theorem, and the fact that the dual space of the space separates points in the space. Below are some common topological vector spaces, roughly in order of increasing "niceness." *
F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex. ...
s are
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
topological vector spaces with a translation-invariant metric. These include L^p spaces for all p > 0. *
Locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
s: here each point has a
local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
consisting of
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
s. By a technique known as
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, then ...
s it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms. Local convexity is the minimum requirement for "geometrical" arguments like the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
. The L^p spaces are locally convex (in fact, Banach spaces) for all p \geq 1, but not for 0 < p < 1. *
Barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a b ...
s: locally convex spaces where the
Banach–Steinhaus theorem In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
holds. *
Bornological space In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a ...
: a locally convex space where the
continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear o ...
s to any locally convex space are exactly the
bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vect ...
s. *
Stereotype space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an iso ...
: a locally convex space satisfying a variant of reflexivity condition, where the dual space is endowed with the topology of uniform convergence on totally bounded sets. *
Montel space In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector ...
: a barrelled space where every
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
and
bounded set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mat ...
is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
*
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
s: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class -- C^\infty(\R) is a Fréchet space under the seminorms \, f\, _ = \sup_ , f^(x), . A locally convex F-space is a Fréchet space. *
LF-space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. This means that ''X'' is a direct limit ...
s are
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
s.
ILH space ILH may refer to: *Interscholastic League of Honolulu, a group of high school teams from Honolulu *International League of Humanists International League of Humanists (ILH) is a non-profit international association of eminent humanists. Its hea ...
s are
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
s of Hilbert spaces. *
Nuclear space In mathematics, nuclear spaces are topological vector space, topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite diff ...
s: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a
nuclear operator In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector space ...
. *
Normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s and
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
s: locally convex spaces where the topology can be described by a single
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
or
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
. In normed spaces a linear operator is continuous if and only if it is bounded. *
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s: Complete
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s. Most of functional analysis is formulated for Banach spaces. This class includes the L^p spaces with 1\leq p \leq \infty, the space BV of functions of bounded variation, and certain spaces of measures. *
Reflexive Banach space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isom ...
s: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is reflexive is L^1, whose dual is L^ but is strictly contained in the dual of L^. *
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s: these have an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include L^2 spaces, the L^2
Sobolev spaces In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
W^, and
Hardy spaces In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . ...
. *
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s: \R^n or \Complex^n with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite n, there is only one n-dimensional topological vector space, up to isomorphism. It follows from this that any finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).


Dual space

Every topological vector space has a
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
—the set X' of all continuous linear functionals, that is,
continuous linear map In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a Continuous function (topology), continuous linear transformation between topological vector spaces. An operator between two norm ...
s from the space into the base field \mathbb. A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation X' \to \mathbb is continuous. This turns the dual into a locally convex topological vector space. This topology is called the
weak-* topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
. This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see
Banach–Alaoglu theorem In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proo ...
). Caution: Whenever X is a non-normable locally convex space, then the pairing map X' \times X \to \mathbb is never continuous, no matter which vector space topology one chooses on X'. A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.


Properties

For any S \subseteq X of a TVS X, the ''convex'' (resp. ''
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
, disked, closed convex, closed balanced, closed disked) ''hull'' of S is the smallest subset of X that has this property and contains S. The closure (respectively, interior,
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
, balanced hull, disked hull) of a set S is sometimes denoted by \operatorname_X S (respectively, \operatorname_X S, \operatorname S, \operatorname S, \operatorname S). The
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
\operatorname S of a subset S is equal to the set of all of elements in S, which are finite linear combinations of the form t_1 s_1 + \cdots + t_n s_n where n \geq 1 is an integer, s_1, \ldots, s_n \in S and t_1, \ldots, t_n \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> sum to 1. The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it.


Neighborhoods and open sets

Properties of neighborhoods and open sets Every TVS is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
and
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness a ...
and any connected open subset of a TVS is arcwise connected. If S \subseteq X and U is an open subset of X then S + U is an open set in X and if S \subseteq X has non-empty interior then S - S is a neighborhood of the origin. The open convex subsets of a TVS X (not necessarily Hausdorff or locally convex) are exactly those that are of the form z + \ ~=~ \ for some z \in X and some positive continuous
sublinear functional In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
p on X. If K is an absorbing
disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...
in a TVS X and if p := p_K is the
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, then ...
of K then \operatorname_X K ~\subseteq~ \ ~\subseteq~ K ~\subseteq~ \ ~\subseteq~ \operatorname_X K where importantly, it was assumed that K had any topological properties nor that p was continuous (which happens if and only if K is a neighborhood of the origin). Let \tau and \nu be two vector topologies on X. Then \tau \subseteq \nu if and only if whenever a net x_ = \left(x_i\right)_ in X converges 0 in (X, \nu) then x_ \to 0 in (X, \tau). Let \mathcal be a neighborhood basis of the origin in X, let S \subseteq X, and let x \in X. Then x \in \operatorname_X S if and only if there exists a net s_ = \left(s_N\right)_ in S (indexed by \mathcal) such that s_ \to x in X. This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets. If X is a TVS that is of the
second category In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
in itself (that is, a
nonmeager space In the Mathematics, mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or Negligible set, negligible in a precise sense detailed below. A set ...
) then any closed convex absorbing subset of X is a neighborhood of the origin. This is no longer guaranteed if the set is not convex (a counter-example exists even in X = \R^2) or if X is not of the second category in itself. Interior If R, S \subseteq X and S has non-empty interior then \operatorname_X S ~=~ \operatorname_X \left(\operatorname_X S\right)~ \text ~\operatorname_X S ~=~ \operatorname_X \left(\operatorname_X S\right) and \operatorname_X (R) + \operatorname_X (S) ~\subseteq~ R + \operatorname_X S \subseteq \operatorname_X (R + S). The
topological interior In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the ...
of a
disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...
is not empty if and only if this interior contains the origin. More generally, if S is a
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
set with non-empty interior \operatorname_X S \neq \varnothing in a TVS X then \ \cup \operatorname_X S will necessarily be balanced; consequently, \operatorname_X S will be balanced if and only if it contains the origin.This is because every non-empty balanced set must contain the origin and because 0 \in \operatorname_X S if and only if \operatorname_X S = \ \cup \operatorname_X S. For this (i.e. 0 \in \operatorname_X S) to be true, it suffices for S to also be convex (in addition to being balanced and having non-empty interior).; The conclusion 0 \in \operatorname_X S could be false if S is not also convex; for example, in X := \R^2, the interior of the closed and balanced set S := \ is \. If C is convex and 0 < t \leq 1, then t \operatorname C + (1 - t) \operatorname C ~\subseteq~ \operatorname C. Explicitly, this means that if C is a convex subset of a TVS X (not necessarily Hausdorff or locally convex), y \in \operatorname_X C, and x \in \operatorname_X C then the open line segment joining x and y belongs to the interior of C; that is, \ \subseteq \operatorname_X C.Fix 0 < r < 1 so it remains to show that w_0 ~\stackrel~ r x + (1 - r) y belongs to \operatorname_X C. By replacing C, x, y with C - w_0, x - w_0, y - w_0 if necessary, we may assume without loss of generality that r x + (1 - r) y = 0, and so it remains to show that C is a neighborhood of the origin. Let s ~\stackrel~ \tfrac < 0 so that y = \tfrac x = s x. Since scalar multiplication by s \neq 0 is a linear homeomorphism X \to X, \operatorname_X \left(\tfrac C\right) = \tfrac \operatorname_X C. Since x \in \operatorname C and y \in \operatorname C, it follows that x = \tfrac y \in \operatorname \left(\tfrac C\right) \cap \operatorname C where because \operatorname C is open, there exists some c_0 \in \left(\tfrac C\right) \cap \operatorname C, which satisfies s c_0 \in C. Define h : X \to X by x \mapsto r x + (1 - r) s c_0 = r x - r c_0, which is a homeomorphism because 0 < r < 1. The set h\left(\operatorname C\right) is thus an open subset of X that moreover contains h(c_0) = r c_0 - r c_0 = 0. If c \in \operatorname C then h(c) = r c + (1 - r) s c_0 \in C since C is convex, 0 < r < 1, and s c_0, c \in C, which proves that h\left(\operatorname C\right) \subseteq C. Thus h\left(\operatorname C\right) is an open subset of X that contains the origin and is contained in C. Q.E.D. If N \subseteq X is any balanced neighborhood of the origin in X then \operatorname_X N \subseteq B_1 N = \bigcup_ a N \subseteq N where B_1 is the set of all scalars a such that , a, < 1. If x belongs to the interior of a convex set S \subseteq X and y \in \operatorname_X S, then the half-open line segment [x, y) := \ \subseteq \operatorname_X \text x \neq y and [x, x) = \varnothing \text x = y. If N is a
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
neighborhood of 0 in X and B_1 := \, then by considering intersections of the form N \cap \R x (which are convex
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
neighborhoods of 0 in the real TVS \R x) it follows that: \operatorname N = [0, 1) \operatorname N = (-1, 1) N = B_1 N, and furthermore, if x \in \operatorname N \text r := \sup \ then r > 1 \text [0, r) x \subseteq \operatorname N, and if r \neq \infty then r x \in \operatorname N \setminus \operatorname N.


Non-Hausdorff spaces and the closure of the origin

A topological vector space X is Hausdorff if and only if \ is a closed subset of X, or equivalently, if and only if \ = \operatorname_X \. Because \ is a vector subspace of X, the same is true of its closure \operatorname_X \, which is referred to as in X. This vector space satisfies \operatorname_X \ = \bigcap_ N so that in particular, every neighborhood of the origin in X contains the vector space \operatorname_X \ as a subset. The
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
on \operatorname_X \ is always the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
, which in particular implies that the topological vector space \operatorname_X \ a compact space (even if its dimension is non-zero or even infinite) and consequently also a Bounded set (topological vector space), bounded subset of X. In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of \. Every subset of \operatorname_X \ also carries the trivial topology and so is itself a compact, and thus also complete, subspace (see footnote for a proof).Since \operatorname_X \ has the trivial topology, so does each of its subsets, which makes them all compact. It is known that a subset of any uniform space is compact if and only if it is complete and totally bounded. In particular, if X is not Hausdorff then there exist subsets that are both but in X; for instance, this will be true of any non-empty proper subset of \operatorname_X \. If S \subseteq X is compact, then \operatorname_X S = S + \operatorname_X \ and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are
relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since ...
), which is not guaranteed for arbitrary non-Hausdorff
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s.In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the
particular point topology In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collecti ...
on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. S + \operatorname_X \ is compact because it is the image of the compact set S \times \operatorname_X \ under the continuous addition map \cdot\, + \,\cdot\; : X \times X \to X. Recall also that the sum of a compact set (that is, S) and a closed set is closed so S + \operatorname_X \ is closed in X.
For every subset S \subseteq X, S + \operatorname_X \ \subseteq \operatorname_X S and consequently, if S \subseteq X is open or closed in X then S + \operatorname_X \ = SIf s \in S then s + \operatorname_X \ = \operatorname_X (s + \) = \operatorname_X \ \subseteq \operatorname_X S. Because S \subseteq S + \operatorname_X \ \subseteq \operatorname_X S, if S is closed then equality holds. Using the fact that \operatorname_X \ is a vector space, it is readily verified that the complement in X of any set S satisfying the equality S + \operatorname_X \ = S must also satisfy this equality (when X \setminus S is substituted for S). (so that this open closed subsets S can be described as a "tube" whose vertical side is the vector space \operatorname_X \). For any subset S \subseteq X of this TVS X, the following are equivalent: * S is
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
. * S + \operatorname_X \ is totally bounded. * \operatorname_X S is totally bounded. * The image if S under the canonical quotient map X \to X / \operatorname_X (\) is totally bounded. If M is a vector subspace of a TVS X then X / M is Hausdorff if and only if M is closed in X. Moreover, the
quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
q : X \to X / \operatorname_X \ is always a
closed map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...
onto the (necessarily) Hausdorff TVS. Every vector subspace of X that is an algebraic complement of \operatorname_X \ (that is, a vector subspace H that satisfies \ = H \cap \operatorname_X \ and X = H + \operatorname_X \) is a
topological complement In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space X, is a vector subspace M for which there exists some other vector subspace N of X, called its (topological) complement in X, such that ...
of \operatorname_X \. Consequently, if H is an algebraic complement of \operatorname_X \ in X then the addition map H \times \operatorname_X \ \to X, defined by (h, n) \mapsto h + n is a TVS-isomorphism, where H is necessarily Hausdorff and \operatorname_X \ has the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
. Moreover, if C is a Hausdorff completion of H then C \times \operatorname_X \ is a completion of X \cong H \times \operatorname_X \.


Closed and compact sets

Compact and totally bounded sets A subset of a TVS is compact if and only if it is complete and
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
. Thus, in a
complete topological vector space In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
, a closed and totally bounded subset is compact. A subset S of a TVS X is
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
if and only if \operatorname_X S is totally bounded, if and only if its image under the canonical quotient map X \to X / \operatorname_X (\) is totally bounded. Every relatively compact set is totally bounded and the closure of a totally bounded set is totally bounded. The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. If S is a subset of a TVS X such that every sequence in S has a cluster point in S then S is totally bounded. If K is a compact subset of a TVS X and U is an open subset of X containing K, then there exists a neighborhood N of 0 such that K + N \subseteq U. Closure and closed set The closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a
barrel A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
. The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. If M is a vector subspace of X and N is a closed neighborhood of the origin in X such that U \cap N is closed in X then M is closed in X. The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see this footnoteIn \R^2, the sum of the y-axis and the graph of y = \frac, which is the complement of the y-axis, is open in \R^2. In \R, the
Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...
\Z + \sqrt\Z is a countable dense subset of \R so not closed in \R.
for examples). If S \subseteq X and a is a scalar then a \operatorname_X S \subseteq \operatorname_X (a S), where if X is Hausdorff, a \neq 0, \text S = \varnothing then equality holds: \operatorname_X (a S) = a \operatorname_X S. In particular, every non-zero scalar multiple of a closed set is closed. If S \subseteq X and if A is a set of scalars such that neither \operatorname S \text \operatorname A contain zero then \left(\operatorname A\right) \left(\operatorname_X S\right) = \operatorname_X (A S). If S \subseteq X \text S + S \subseteq 2 \operatorname_X S then \operatorname_X S is convex. If R, S \subseteq X then \operatorname_X (R) + \operatorname_X (S) ~\subseteq~ \operatorname_X (R + S)~ \text ~\operatorname_X \left \operatorname_X (R) + \operatorname_X (S) \right~=~ \operatorname_X (R + S) and so consequently, if R + S is closed then so is \operatorname_X (R) + \operatorname_X (S). If X is a real TVS and S \subseteq X, then \bigcap_ r S \subseteq \operatorname_X S where the left hand side is independent of the topology on X; moreover, if S is a convex neighborhood of the origin then equality holds. For any subset S \subseteq X, \operatorname_X S ~=~ \bigcap_ (S + N) where \mathcal is any neighborhood basis at the origin for X. However, \operatorname_X U ~\supseteq~ \bigcap \ and it is possible for this containment to be proper (for example, if X = \R and S is the rational numbers). It follows that \operatorname_X U \subseteq U + U for every neighborhood U of the origin in X. Closed hulls In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general. * The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to \operatorname_X (\operatorname S). * The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to \operatorname_X (\operatorname S). * The closed disked hull of a set is equal to the closure of the disked hull of that set; that is, equal to \operatorname_X (\operatorname S). If R, S \subseteq X and the closed convex hull of one of the sets S or R is compact then \operatorname_X (\operatorname (R + S)) ~=~ \operatorname_X (\operatorname R) + \operatorname_X (\operatorname S). If R, S \subseteq X each have a closed convex hull that is compact (that is, \operatorname_X (\operatorname R) and \operatorname_X (\operatorname S) are compact) then \operatorname_X (\operatorname (R \cup S)) ~=~ \operatorname \left \operatorname_X (\operatorname R) \cup \operatorname_X (\operatorname S) \right Hulls and compactness In a general TVS, the closed convex hull of a compact set may to be compact. The balanced hull of a compact (respectively,
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
) set has that same property. The convex hull of a finite union of compact sets is again compact and convex.


Other properties

Meager, nowhere dense, and Baire A
disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...
in a TVS is not
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
if and only if its closure is a neighborhood of the origin. A vector subspace of a TVS that is closed but not open is
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
. Suppose X is a TVS that does not carry the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
. Then X is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
if and only if X has no balanced absorbing nowhere dense subset. A TVS X is a Baire space if and only if X is
nonmeager In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
, which happens if and only if there does not exist a
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
set D such that X = \bigcup_ n D. Every
nonmeager In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
locally convex TVS is a
barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a b ...
. Important algebraic facts and common misconceptions If S \subseteq X then 2 S \subseteq S + S; if S is convex then equality holds. For an example where equality does hold, let x be non-zero and set S = \; S = \ also works. A subset C is convex if and only if (s + t) C = s C + t C for all positive real s > 0 \text t > 0, or equivalently, if and only if t C + (1 - t) C \subseteq C for all 0 \leq t \leq 1. The
convex balanced hull In mathematics, a subset ''C'' of a Real number, real or Complex number, complex vector space is said to be absolutely convex or disked if it is Convex set, convex and Balanced set, balanced (some people use the term "circled" instead of "balanced") ...
of a set S \subseteq X is equal to the convex hull of the
balanced hull In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
of S; that is, it is equal to \operatorname (\operatorname S). But in general, \operatorname (\operatorname S) ~\subseteq~ \operatorname S ~=~ \operatorname (\operatorname S), where the inclusion might be strict since the
balanced hull In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
of a convex set need not be convex (counter-examples exist even in \R^2). If R, S \subseteq X and a is a scalar then a(R + S) = aR + a S,~ \text ~\operatorname (R + S) = \operatorname R + \operatorname S,~ \text ~\operatorname (a S) = a \operatorname S. If R, S \subseteq X are convex non-empty disjoint sets and x \not\in R \cup S, then S \cap \operatorname (R \cup \) = \varnothing or R \cap \operatorname (S \cup \) = \varnothing. In any non-trivial vector space X, there exist two disjoint non-empty convex subsets whose union is X. Other properties Every TVS topology can be generated by a of ''F''-seminorms. If P(x) is some unary
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
(a true or false statement dependent on x \in X) then for any z \in X, z + \ = \.z + \ = \ = \ and so using y = z + x and the fact that z + X = X, this is equal to \ = \ = \.
Q.E.D. Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...
\blacksquare
So for example, if P(x) denotes "\, x\, < 1" then for any z \in X, z + \ = \. Similarly, if s \neq 0 is a scalar then s \ = \left\. The elements x \in X of these sets must range over a vector space (that is, over X) rather than not just a subset or else these equalities are no longer guaranteed; similarly, z must belong to this vector space (that is, z \in X).


Properties preserved by set operators

* The balanced hull of a compact (respectively,
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
, open) set has that same property. * The (Minkowski) sum of two compact (respectively, bounded, balanced, convex) sets has that same property. But the sum of two closed sets need be closed. * The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need be closed. And the convex hull of a bounded set need be bounded. The following table, the color of each cell indicates whether or not a given property of subsets of X (indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red. So for instance, since the union of two absorbing sets is again absorbing, the cell in row "R \cup S" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in.


See also

* * * * * * * * * * * * * * * *


Notes


Proofs


Citations


Bibliography

* * * * * * * * *


Further reading

* * * * * * * * * * * * *


External links

* {{Authority control Articles containing proofs Topology of function spaces Topological spaces Vector spaces