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, norm, topology, etc.) and the linear functions defi ...
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 ...
, a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
in 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 ...
is called bounded or von Neumann bounded, if every
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
zero vector
In mathematics, a zero element is one of several generalizations of 0, 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 iden ...
can be ''inflated'' to include the set.
A set that is not bounded is called unbounded.
Bounded sets are a natural way to define
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 v ...
polar topologies on the
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 ...
s in a
dual pair, as the
polar set
In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X lying in the dual space X^.
The bipolar of a subset is the polar of A^, but lies ...
of a bounded set is an
absolutely convex and
absorbing set.
The concept was first introduced by
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 c ...
and
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
in
1935
Events
January
* January 7 – Italian premier Benito Mussolini and French Foreign Minister Pierre Laval conclude an agreement, in which each power agrees not to oppose the other's colonial claims.
* January 12 – Amelia Earhart ...
.
Definition
Suppose
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 ...
(TVS) over a
field
A subset
of
is called or just in
if any of the following equivalent conditions are satisfied:
- : For every neighborhood of the origin there exists a real such that
[For any set and scalar the notation is denotes the set ] for all scalars satisfying
* This was the definition introduced by 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 c ...
in 1935.
- is absorbed by every
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.
- For every neighborhood of the origin there exists a scalar such that
- For every neighborhood of the origin there exists a real such that for all scalars satisfying
- For every neighborhood of the origin there exists a real such that for all real
- Any one of statements (1) through (5) above but with the word "neighborhood" replaced by any of the following: " balanced neighborhood," "open balanced neighborhood," "closed balanced neighborhood," "open neighborhood," "closed neighborhood".
* e.g. Statement (2) may become: is bounded if and only if is absorbed by every balanced neighborhood of the origin.
* If 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 v ...
then the adjective "convex" may be also be added to any of these 5 replacements.
- For every sequence of scalars that converges to and every sequence in the sequence converges to in
* This was the definition of "bounded" that
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
used in 1934, which is the same as the definition introduced by Stanisław Mazur
Stanisław Mieczysław Mazur (1 January 1905, Lwów – 5 November 1981, Warsaw) was a Polish mathematician and a member of the Polish Academy of Sciences.
Mazur made important contributions to geometrical methods in linear and nonlinear functio ...
and Władysław Orlicz
Władysław Roman Orlicz (May 24, 1903 in Okocim, Austria-Hungary (now Poland) – August 9, 1990 in Poznań, Poland) was a Polish mathematician of Lwów School of Mathematics. His main interests were functional analysis and topology: Orlicz sp ...
in 1933 for metrizable TVS. Kolmogorov used this definition to prove that a TVS is seminormable if and only if it has a bounded convex neighborhood of the origin.
- For every sequence in the sequence converges to in
- Every
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
subset of is bounded (according to any defining condition other than this one).
If
is a
neighborhood basis for
at the origin then this list may be extended to include:
- Any one of statements (1) through (5) above but with the neighborhoods limited to those belonging to
* e.g. Statement (3) may become: For every there exists a scalar such that
If
is a
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 v ...
space whose topology is defined by a family
of continuous
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 a ...
s, then this list may be extended to include:
- is bounded for all
- There exists a sequence of non-zero scalars such that for every sequence in the sequence is bounded in (according to any defining condition other than this one).
- For all is bounded (according to any defining condition other than this one) in the semi normed space
If
is a
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 ...
with
norm (or more generally, if it is a
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 ...
and
is merely 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 a ...
),
[This means that the topology on is equal to the topology induced on it by Note that every ]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 ...
is a seminormed space and every norm is a seminorm. The definition of the topology induced by a seminorm is identical to the definition of the topology induced by a norm. then this list may be extended to include:
- is a ''norm bounded'' subset of By definition, this means that there exists a real number such that for all
-
* Thus, if is 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 mapping V \to W between two vector spaces that ...
between two normed (or seminormed) spaces and if is the closed (alternatively, open) unit ball in centered at the origin, then is a 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 ...
(which recall means that its operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introd ...
is finite) if and only if the image of this ball under is a norm bounded subset of
- is a subset of some (open or closed) ball.
[If is a ]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 ...
or a 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 ...
, then the open and closed balls of radius (where is a real number) centered at a point are, respectively, the sets and Any such set is call a (non-degenerate) .
* This ball need not be centered at the origin, but its radius must (as usual) be positive and finite.
If
is a vector subspace of the TVS
then this list may be extended to include:
- is contained in the closure of
* In other words, a vector subspace of is bounded if and only if it is a subset of (the vector space)
* Recall that 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 ma ...
if and only if is closed in So the only bounded vector subspace of a Hausdorff TVS is
A subset that is not bounded is called .
Bornology and fundamental systems of bounded sets
The collection of all bounded sets on a topological vector space
is called the or the ()
A or of
is a set
of bounded subsets of
such that every bounded subset of
is a subset of some
The set of all bounded subsets of
trivially forms a fundamental system of bounded sets of
Examples
In any
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 v ...
TVS, the set of closed and bounded
disks are a base of bounded set.
Examples and sufficient conditions
Unless indicated otherwise, 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) need not be
Hausdorff nor
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 v ...
.
- Finite sets are bounded.
- Every totally bounded subset of a TVS is bounded.
- Every
relatively compact set
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 (sinc ...
in a topological vector space is bounded. If the space is equipped with 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 ...
the converse is also true.
- The set of points of a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
is bounded, the set of points of a Cauchy net need not be bounded.
- The closure of the origin (referring to the closure of the set ) is always a bounded closed vector subspace. This set is the unique largest (with respect to set inclusion ) bounded vector subspace of In particular, if is a bounded subset of then so is
Unbounded sets
A set that is not bounded is said to be ''unbounded''.
Any vector subspace of a TVS that is not a contained in the closure of
is unbounded
There exists 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 ...
having a bounded subset
and also a dense vector subspace
such that
is contained in the closure (in
) of any bounded subset of
Stability properties
- In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures,
interiors
''Interiors'' is a 1978 American drama film written and directed by Woody Allen. It stars Kristin Griffith, Mary Beth Hurt, Richard Jordan, Diane Keaton, E. G. Marshall, Geraldine Page, Maureen Stapleton, and Sam Waterston.
Allen's first ful ...
, and 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, \l ...
s of bounded sets are again bounded.
- In any locally convex 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 ...
(also called the convex envelope) of a bounded set is again bounded. However, this may be false if the space is not locally convex, as the (non-locally convex) Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
spaces for have no nontrivial open convex subsets.
- The image of a bounded set under a continuous linear map is a bounded subset of the codomain.
- A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded.
- If and is a topological vector subspace of then is bounded in if and only if is bounded in
* In other words, a subset is bounded in if and only if it is bounded in every (or equivalently, in some) topological vector superspace of
Properties
A
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 ...
has a bounded neighborhood of zero
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 bic ...
its topology can be defined by 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 a ...
.
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 cli ...
of a bounded set is an
absolutely convex and
absorbing set.
Using the definition of
uniformly bounded sets given below,
Mackey's countability condition can be restated as: If
are bounded subsets of a
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) s ...
locally convex 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 v ...
then there exists a sequence
of positive real numbers such that
are
uniformly bounded.
In words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its own positive real so that they become uniformly bounded.
Generalizations
Uniformly bounded sets
A
family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fami ...
of 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 ...
is said to be in
if there exists some bounded subset
of
such that
which happens if and only if its union
is a bounded subset of
In the case of a
normed (or
seminormed) space, a family
is uniformly bounded if and only if its union
is ''norm bounded'', meaning that there exists some real
such that
for every
or equivalently, if and only if
A set
of maps from
to
is said to be
if the family
is uniformly bounded in
which by definition means that there exists some bounded subset
of
such that
or equivalently, if and only if
is a bounded subset of
A set
of linear maps between two normed (or seminormed) spaces
and
is uniformly bounded on some (or equivalently, every) open ball (and/or non-degenerate closed ball) in
if and only if their
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introd ...
s are uniformly bounded; that is, if and only if
Assume
is equicontinuous and let
be a neighborhood of the origin in
Since
is equicontinuous, there exists a neighborhood
of the origin in
such that
for every
Because
is bounded in
there exists some real
such that if
then
So for every
and every
which implies that
Thus
is bounded in
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 ...
Let
be a
balanced neighborhood of the origin in
and let
be a closed balanced neighborhood of the origin in
such that
Define
which is a closed subset of
(since
is closed while every
is continuous) that satisfies
for every
Note that for every non-zero scalar
the set
is closed in
(since scalar multiplication by
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 isom ...
) and so every
is closed in
It will now be shown that
from which
follows.
If
then
being bounded guarantees the existence of some positive integer
such that
where the linearity of every
now implies
thus
and hence
as desired.
Thus
expresses
as a countable union of closed (in
) sets.
Since
is a
nonmeager subset of itself (as it 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 e ...
by the
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
), this is only possible if there is some integer
such that
has non-empty interior in
Let
be any point belonging to this open subset of
Let
be any balanced open neighborhood of the origin in
such that
The sets
form an increasing (meaning
implies
) cover of the compact space
so there exists some
such that
(and thus
).
It will be shown that
for every
thus demonstrating that
is uniformly bounded in
and completing the proof.
So fix
and
Let
The convexity of
guarantees
and moreover,
since
Thus
which is a subset of
Since
is balanced and
we have
which combined with
gives
Finally,
and
imply
as desired.
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 ...
Since every
singleton subset of
is also a bounded subset, it follows that if
is an
equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
set of
continuous linear operators between two
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
and
(not necessarily
Hausdorff or locally convex), then the
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of every
is a bounded subset of
Bounded subsets of topological modules
The definition of bounded sets can be generalized to
topological module In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Examples
A topological vector space is a topological module over a topological field.
An abelian topological ...
s.
A subset
of a topological module
over a
topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps:
R \times R \to R
where R \times R carries the product topology. That means R is an additive ...
is bounded if for any neighborhood
of
there exists a neighborhood
of
such that
See also
*
*
*
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*
*
*
*
*
References
Notes
Bibliography
*
*
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{{Topological vector spaces
Topological vector spaces