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 ...
and related areas of
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 ...
, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of
topological vector space
In 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.
A topological vector space is a vector space that is als ...
s (TVS) that generalize
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. They can be defined as
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
vector spaces whose topology is
generated by translations of
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 ...
,
absorbent,
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. Alternatively they can be defined as 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 ...
with a
family
Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of
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 ...
s, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily
normable, the existence of a convex
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 ...
for the
zero vector
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
is strong enough for 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 ...
to hold, yielding a sufficiently rich theory of continuous
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 ...
s.
Fréchet spaces are locally convex spaces that are
completely metrizable In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' ind ...
(with a choice of complete metric). They are generalizations of
Banach spaces
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 ...
, which are complete vector spaces with respect to a metric generated by a
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 ...
.
History
Metrizable topologies on vector spaces have been studied since their introduction in
Maurice Fréchet's 1902 PhD thesis ''Sur quelques points du calcul fonctionnel'' (wherein the notion of 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 mathem ...
was first introduced).
After the notion of a general topological space was defined by
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and ...
in 1914, although locally convex topologies were implicitly used by some mathematicians, up to 1934 only
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
would seem to have explicitly defined 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 ...
on Hilbert spaces and
strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
on operators on Hilbert spaces. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a ''convex space'' by him).
A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like
nets, 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 ...
and
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
) to be proven in its full generality, is the
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 ...
which
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
first established in 1932 by an elementary diagonal argument for the case of separable normed spaces (in which case the
unit ball of the dual is metrizable).
Definition
Suppose
is a vector space over
a
subfield of the
complex numbers
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 a ...
(normally
itself or
).
A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.
Definition via convex sets
A subset
in
is called
#
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 for all
and
In other words,
contains all line segments between points in
#
Circled if for all
and scalars
if
then
If
this means that
is equal to its reflection through the origin. For
it means for any
contains the circle through
centred on the origin, in the one-dimensional complex subspace generated by
#
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 ...
if for all
and scalars
if
then
If
this means that if
then
contains the line segment between
and
For
it means for any in
contains the disk with
on its boundary, centred on the origin, in the one-dimensional complex subspace generated by
Equivalently, a balanced set is a circled cone.
# A
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
(when the underlying
field is ordered) if for all
and
#
Absorbent or absorbing if for every
there exists
such that
for all
satisfying
The set
can be scaled out by any "large" value to absorb every point in the space.
#* In any TVS, every neighborhood of the origin is absorbent.
#
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 ...
or a disk if it is both balanced and convex. This is equivalent to it being closed under linear combinations whose coefficients absolutely sum to
; such a set is absorbent if it spans all of
A
topological vector space
In 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.
A topological vector space is a vector space that is als ...
(TVS) is called locally convex if the origin has 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 ...
(that is, a local base) consisting of convex sets.
In fact, every locally convex TVS has a neighborhood basis of the origin consisting of sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets.
Every TVS has a neighborhood basis at the origin consisting of balanced sets but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced convex. It is possible for a TVS to have neighborhoods of the origin that are convex and yet be locally convex.
Because translation is (by definition of "topological vector space") continuous, all translations are
homeomorphisms
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 isomorp ...
, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.
Definition via seminorms
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 ...
on
is a map
such that
#
is nonnegative or positive semidefinite:
;
#
is positive homogeneous or positive scalable:
for every scalar
So, in particular,
;
#
is subadditive. It satisfies the triangle inequality:
If
satisfies positive definiteness, which states that if
then
then
is a
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 ...
.
While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.
If
is a vector space and
is a family of seminorms on
then a subset
of
is called a base of seminorms for
if for all
there exists a
and a real
such that
Definition (second version): A locally convex space is defined to be a vector space
along with a
family
Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of seminorms on
Seminorm topology
Suppose that
is a vector space over
where
is either the real or complex numbers.
A family of seminorms
on the vector space
induces a canonical vector space topology on
, called the
initial topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' tha ...
induced by the seminorms, making it into a
topological vector space
In 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.
A topological vector space is a vector space that is als ...
(TVS). By definition, it is the
coarsest topology on
for which all maps in
are continuous.
That the vector space operations are continuous in this topology follows from properties 2 and 3 above.
It is possible for a locally convex topology on a space
to be induced by a family of norms but for
to be
normable (that is, to have its topology be induced by a single norm).
=Basis and subbases
=
Let
denote the open ball of radius
in
. The family of sets
as
ranges over a family of seminorms
and
ranges over the positive real numbers
is a
subbasis at the origin for the topology induced by
. These sets are convex, as follows from properties 2 and 3 of seminorms.
Intersections of finitely many such sets are then also convex, and since the collection of all such finite intersections is a
basis at the origin it follows that the topology is locally convex in the sense of the definition given above.
Recall that the topology of a TVS is translation invariant, meaning that if
is any subset of
containing the origin then for any
is a neighborhood of the origin if and only if
is a neighborhood of
;
thus it suffices to define the topology at the origin.
A base of neighborhoods of
for this topology is obtained in the following way: for every finite subset
of
and every
let
=Bases of seminorms and saturated families
=
If
is a locally convex space and if
is a collection of continuous seminorms on
, then
is called a base of continuous seminorms if it is a base of seminorms for the collection of continuous seminorms on
. Explicitly, this means that for all continuous seminorms
on
, there exists a
and a real
such that
If
is a base of continuous seminorms for a locally convex TVS
then the family of all sets of the form
as
varies over
and
varies over the positive real numbers, is a of neighborhoods of the origin in
(not just a subbasis, so there is no need to take finite intersections of such sets).
[Let be the open unit ball associated with the seminorm and note that if is real then and so Thus a basic open neighborhood of the origin induced by is a finite intersection of the form where and are all positive reals. Let which is a continuous seminorm and moreover, Pick and such that where this inequality holds if and only if Thus as desired.]
A family
of seminorms on a vector space
is called saturated if for any
and
in
the seminorm defined by
belongs to
If
is a saturated family of continuous seminorms that induces the topology on
then the collection of all sets of the form
as
ranges over
and
ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets;
This forms a basis at the origin rather than merely a subbasis so that in particular, there is need to take finite intersections of such sets.
Basis of norms
The following theorem implies that if
is a locally convex space then the topology of
can be a defined by a family of continuous on
(a
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 ...
is 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 ...
where
implies
) if and only if there exists continuous on
. This is because the sum of a norm and a seminorm is a norm so if a locally convex space is defined by some family
of seminorms (each of which is necessarily continuous) then the family
of (also continuous) norms obtained by adding some given continuous norm
to each element, will necessarily be a family of norms that defines this same locally convex topology.
If there exists a continuous norm on a topological vector space
then
is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or
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 th ...
s).
=Nets
=
Suppose that the topology of a locally convex space
is induced by a family
of continuous seminorms on
.
If
and if
is 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 ...
in
, then
in
if and only if for all
Moreover, if
is Cauchy in
, then so is
for every
Equivalence of definitions
Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice.
The equivalence of the two definitions follows from a construction known as the
Minkowski functional or Minkowski gauge.
The key feature of seminorms which ensures the convexity of their
-
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
s is 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 ...
.
For an absorbing set
such that if
then
whenever
define the Minkowski functional of
to be
From this definition it follows that
is a seminorm if
is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets
form a base of convex absorbent balanced sets.
Ways of defining a locally convex topology
Example: auxiliary normed spaces
If
is
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 ...
and
absorbing in
then the
symmetric set
In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements.
Definition
In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs to ...
will be convex and
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 ...
(also known as an or a ) in addition to being absorbing in
This guarantees that the
Minkowski functional of
will be 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 ...
on
thereby making
into a
seminormed space that carries its canonical
pseduometrizable topology. The set of scalar multiples
as
ranges over
(or over any other set of non-zero scalars having
as a limit point) forms a neighborhood basis of absorbing
disks at the origin for this locally convex topology. If
is a
topological vector space
In 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.
A topological vector space is a vector space that is als ...
and if this convex absorbing subset
is also a
bounded subset of
then the absorbing disk
will also be bounded, in which case
will be a
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 ...
and
will form what is known as an
auxiliary normed space. If this normed space is a
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 ...
then
is called a .
Further definitions
* A family of seminorms
is called total or separated or is said to separate points if whenever
holds for every
then
is necessarily
A locally convex space is
Hausdorff if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.
* A
pseudometric is a generalization of a metric which does not satisfy the condition that
only when
A locally convex space is pseudometrizable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by
(where the
can be replaced by any positive
summable sequence
). This pseudometric is translation-invariant, but not homogeneous, meaning
and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a
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 th ...
.
* As with any topological vector space, a locally convex space is also a
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 ...
. Thus one may speak of
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 ...
,
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
Cauchy sequence
In 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 m ...
s.
* A
Cauchy net
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomai ...
in a locally convex space is 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 ...
such that for every
and every seminorm
there exists some index
such that for all indices
In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar
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 ...
s because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of
pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space 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 ...
if and only if every Cauchy net converges.
* A family of seminorms becomes a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
ed set under the relation
if and only if there exists an
such that for all
One says it is a directed family of seminorms if the family is a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
with addition as the
join Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two top ...
, in other words if for every
and
there is a
such that
Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family
let
be the set of finite subsets of
and then for every
define
One may check that
is an equivalent directed family.
* If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if the origin has a
bounded neighborhood.
Sufficient conditions
Hahn–Banach extension property
Let
be a TVS.
Say that a vector subspace
of
has the extension property if any continuous linear functional on
can be extended to a continuous linear functional on
.
Say that
has the
Hahn-Banach extension property (HBEP) if every vector subspace of
has the extension property.
The
Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.
For complete
metrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
s there is a converse:
If a vector space
has uncountable dimension and if we endow it with the
finest vector topology
In 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.
A topological vector space is a vector space that is al ...
then this is a TVS with the HBEP that is neither locally convex or metrizable.
Properties
Throughout,
is a family of continuous seminorms that generate the topology of
Topological closure
If
and
then
if and only if for every
and every finite collection
there exists some
such that
The closure of
in
is equal to
Topology of Hausdorff locally convex spaces
Every Hausdorff locally convex space is
homeomorphic
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 a vector subspace of a product of
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.
The
Anderson–Kadec theorem
In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. The ...
states that every infinite–dimensional
separable 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 th ...
is
homeomorphic
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 the
product space
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 ...
of countably many copies of
(this homeomorphism need not be a
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 ...
).
Properties of convex subsets
Algebraic properties of convex subsets
A subset
is convex if and only if
for all
or equivalently, if and only if
for all positive real
where because
always holds, the
equals sign
The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between two ...
can be replaced with
If
is a convex set that contains the origin then
is
star shaped at the origin and for all non-negative real
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 ...
of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex.
Topological properties of convex subsets
* Suppose that
is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of
are exactly those that are of the form
for some
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. ...
on
* The interior and closure of a convex subset of a TVS is again convex.
* If
is a convex set with non-empty interior, then the closure of
is equal to the closure of the interior of
; furthermore, the interior of
is equal to the interior of the closure of
** So if the interior of a convex set
is non-empty then
is a closed (respectively, open) set if and only if it is a regular closed (respectively, regular open) set.
* If
is convex and
then
Explicitly, this means that if
is a convex subset of a TVS
(not necessarily Hausdorff or locally convex),
belongs to the closure of
and
belongs to the interior of
then the open line segment joining
and
belongs to the interior of
that is,
[Fix so it remains to show that belongs to By replacing with if necessary, we may assume without loss of generality that and so it remains to show that is a neighborhood of the origin. Let so that Since scalar multiplication by is a linear homeomorphism Since and it follows that where because is open, there exists some which satisfies Define by which is a homeomorphism because The set is thus an open subset of that moreover contains If then since is convex, and which proves that Thus is an open subset of that contains the origin and is contained in Q.E.D.]
* If
is a closed vector subspace of a (not necessarily Hausdorff) locally convex space
is a convex neighborhood of the origin in
and if
is a vector in
then there exists a convex neighborhood
of the origin in
such that
and
* The closure of a convex subset of a locally convex Hausdorff space
is the same for locally convex Hausdorff TVS topologies on
that are compatible with
duality between
and its continuous dual space.
* In a locally convex space, the convex hull and the
disked hull 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 ...
of a totally bounded set is totally bounded.
* In a
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 ...
locally convex space, the convex hull and the disked hull of a compact set are both compact.
** More generally, if
is a compact subset of a locally convex space, then the convex hull
(respectively, the disked hull
) is compact if and only if it is complete.
* In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.
* In a
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 th ...
, the closed convex hull of a compact set is compact.
* In a locally convex space, any linear combination of totally bounded sets is totally bounded.
Properties of convex hulls
For any subset
of a TVS
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 ...
(respectively, closed convex hull,
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, ...
, convex balanced hull) of
denoted by
(respectively,
), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of
containing
* The convex hull of compact subset of a
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 ...
is necessarily closed and so also necessarily compact. For example, let
be the separable Hilbert space
of square-summable sequences with the usual norm
and let
be the standard
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
(that is
at the
-coordinate). The closed set
is compact but its convex hull
is a closed set because
belongs to the closure of
in
but
(since every sequence
is a finite
convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...
of elements of
and so is necessarily
in all but finitely many coordinates, which is not true of
). However, like in all
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 locally convex spaces, the convex hull
of this compact subset is compact. The vector subspace
is a
pre-Hilbert space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often d ...
when endowed with the substructure that the Hilbert space
induces on it but
is not complete and
(since
). The closed convex hull of
in
(here, "closed" means with respect to
and not to
as before) is equal to
which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might to be compact (although it will be
precompact/totally bounded).
* In a Hausdorff locally convex space
the closed convex hull
of compact subset
is not necessarily compact although it is a
precompact (also called "totally bounded") subset, which means that its closure,
of
will be compact (here
so that
if and only if
is complete); that is to say,
will be compact. So for example, the closed convex hull
of a compact subset of
of a
pre-Hilbert space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often d ...
is always a precompact subset of
and so the closure of
in any Hilbert space
containing
(such as the Hausdorff completion of
for instance) will be compact (this is the case in the previous example above).
* In a
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
locally convex TVS, the closure of the convex hull of a compact subset is again compact.
* In a Hausdorff locally convex TVS, the convex hull of a
precompact set is again precompact. Consequently, in a
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 locally convex space, the closed convex hull of a compact subset is again compact.
* In any TVS, the convex hull of a finite union of compact convex sets is compact (and convex).
** This implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is (in addition to being compact and convex); in particular, the convex hull of such a union is equal to the convex hull of that union.
** In general, the closed convex hull of a compact set is not necessarily compact.
** In any non-Hausdorff TVS, there exist subsets that are compact (and thus complete) but closed.
* The
bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the Polar set, polar of the polar) of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions f ...
states that the bipolar (that is, the
polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
* Polar climate, the c ...
of the polar) of a subset of a locally convex Hausdorff TVS is equal to the closed convex balanced hull of that set.
* 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 is necessarily convex.
* If
and
are convex subsets of a
topological vector space
In 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.
A topological vector space is a vector space that is als ...
and if
then there exist
and a real number
satisfying
such that
* If
is a vector subspace of a TVS
a convex subset of
and
a convex subset of
such that
then
* Recall that the smallest
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 ...
subset of
containing a set
is called the balanced hull of
and is denoted by
For any subset
of
the convex balanced hull of
denoted by
is the smallest subset of
containing
that is convex and balanced. The convex balanced hull of
is equal to the convex hull of the balanced hull of
(i.e.
), but the convex balanced hull of
is necessarily equal to the balanced hull of the convex hull of
(that is,
is not necessarily equal to
).
* If
are subsets of a TVS
and if
is a scalar then
and
Moreover, if
is compact then
However, the convex hull of a closed set need not be closed; for example, the set
is closed in
but its convex hull is the open set
* If
are subsets of a TVS
whose closed convex hulls are compact, then
* If
is a convex set in a complex vector space
and there exists some
such that
then
for all real
such that
In particular,
for all scalars
such that
Examples and nonexamples
Finest and coarsest locally convex topology
Coarsest vector topology
Any vector space
endowed with 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 ...
(also called 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 ...
) is a locally convex TVS (and of course, it is the coarsest such topology).
This topology is Hausdorff if and only
The indiscrete topology makes any vector space into a
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 locally convex TVS.
In contrast, 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 ...
forms a vector topology on
if and only
This follows from the fact that every
topological vector space
In 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.
A topological vector space is a vector space that is als ...
is a
connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties tha ...
.
Finest locally convex topology
If
is a real or complex vector space and if
is the set of all seminorms on
then the locally convex TVS topology, denoted by
that
induces on
is called the on
This topology may also be described as the TVS-topology on
having as a neighborhood base at the origin the set of all
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 ...
s in
Any locally convex TVS-topology on
is necessarily a subset of
is
Hausdorff.
Every linear map from
into another locally convex TVS is necessarily continuous.
In particular, every linear functional on
is continuous and every vector subspace of
is closed in
;
therefore, if
is infinite dimensional then
is not pseudometrizable (and thus not metrizable).
Moreover,
is the Hausdorff locally convex topology on
with the property that any linear map from it into any Hausdorff locally convex space is continuous.
The space
is a
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 ...
.
Examples of locally convex spaces
Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalizes parts of the theory of normed spaces.
The family of seminorms can be taken to be the single norm.
Every Banach space is a complete Hausdorff locally convex space, in particular, the
spaces with
are locally convex.
More generally, every Fréchet space is locally convex.
A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.
The space
of
real valued sequences
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural n ...
with the family of seminorms given by
is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. This is also the
limit topology
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 ...
of the spaces
embedded in
in the natural way, by completing finite sequences with infinitely many
Given any vector space
and a collection
of linear functionals on it,
can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in
continuous. This is known as 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 ...
or the
initial topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' tha ...
determined by
The collection
may be the
algebraic dual
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 con ...
of
or any other collection.
The family of seminorms in this case is given by
for all
in
Spaces of differentiable functions give other non-normable examples. Consider the space of
smooth functions
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 ...
such that
where
and
are
multiindices
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
.
The family of seminorms defined by
is separated, and countable, and the space is complete, so this metrizable space is a Fréchet space.
It is known as 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 ...
, or the space of functions of rapid decrease, and its
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 const ...
is the space of
tempered distribution
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.
An important
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
in functional analysis is the space
of smooth functions with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
in
A more detailed construction is needed for the topology of this space because the space
is not complete in the uniform norm. The topology on
is defined as follows: for any fixed
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
the space
of functions
with
is a
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 th ...
with countable family of seminorms
(these are actually norms, and the completion of the space
with the
norm is a Banach space
).
Given any collection
of compact sets, directed by inclusion and such that their union equal
the
form a
direct system
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from ...
, and
is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an
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 limi ...
. More concretely,
is the union of all the
with the strongest topology which makes each
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
continuous.
This space is locally convex and complete. However, it is not metrizable, and so it is not a Fréchet space. The dual space of
is the space of
distributions on
More abstractly, given 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 ...
the space
of continuous (not necessarily bounded) functions on
can be given the topology 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 ...
on compact sets. This topology is defined by semi-norms
(as
varies over the
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
of all compact subsets of
). When
is locally compact (for example, an open set in
) the
Stone–Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of
that separates points and contains the constant functions (for example, the subalgebra of polynomials) is
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 ...
.
Examples of spaces lacking local convexity
Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:
* The
spaces ">, 1 for
are equipped with the
F-norm They are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces
with an atomless, finite measure
and
are not locally convex.
* The space of
measurable
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
functions on the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...