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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 X 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 \mathbb. A subset B of X is called or just in X if any of the following equivalent conditions are satisfied:
  1. : For every neighborhood V of the origin there exists a real r > 0 such that B \subseteq s VFor any set A and scalar s, the notation s A is denotes the set s A := \. for all scalars s satisfying , s, \geq r. * 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.
  2. B 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.
  3. For every neighborhood V of the origin there exists a scalar s such that B \subseteq s V.
  4. For every neighborhood V of the origin there exists a real r > 0 such that s B \subseteq V for all scalars s satisfying , s, \leq r.
  5. For every neighborhood V of the origin there exists a real r > 0 such that t B \subseteq V for all real 0 < t \leq r.
  6. 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: B is bounded if and only if B is absorbed by every balanced neighborhood of the origin. * If X 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.
  7. For every sequence of scalars s_1, s_2, s_3, \ldots that converges to 0 and every sequence b_1, b_2, b_3, \ldots in B, the sequence s_1 b_1, s_2 b_2, s_3 b_3, \ldots converges to 0 in X. * 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.
  8. For every sequence b_1, b_2, b_3, \ldots in B, the sequence \left(\tfrac b_i\right)_^ converges to 0 in X.
  9. 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 B is bounded (according to any defining condition other than this one).
If \mathcal is a neighborhood basis for X at the origin then this list may be extended to include:
  1. Any one of statements (1) through (5) above but with the neighborhoods limited to those belonging to \mathcal. * e.g. Statement (3) may become: For every V \in \mathcal there exists a scalar s such that B \subseteq s V.
If X 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 \mathcal 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:
  1. p(B) is bounded for all p \in \mathcal.
  2. There exists a sequence of non-zero scalars s_1, s_2, s_3, \ldots such that for every sequence b_1, b_2, b_3, \ldots in B, the sequence b_1 s_1, b_2 s_2, b_3 s_3, \ldots is bounded in X (according to any defining condition other than this one).
  3. For all p \in \mathcal, B is bounded (according to any defining condition other than this one) in the semi normed space (X, p).
If X 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 \, \cdot\, (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 \, \cdot\, 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 X is equal to the topology induced on it by \, \cdot\, . 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:
  1. B is a ''norm bounded'' subset of (X, \, \cdot\, ). By definition, this means that there exists a real number r > 0 such that \, b\, \leq r for all b \in B.
  2. \sup_ \, b\, < \infty. * Thus, if L : (X, \, \cdot\, ) \to (Y, \, \cdot\, ) 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 B is the closed (alternatively, open) unit ball in (X, \, \cdot\, ) centered at the origin, then L 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 ...
    \, L\, := \sup_ \, L(b)\, < \infty is finite) if and only if the image L(B) of this ball under L is a norm bounded subset of (Y, \, \cdot\, ).
  3. B is a subset of some (open or closed) ball.If (X, \, \cdot\, ) 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 r > 0 (where r \neq \infty is a real number) centered at a point x \in X are, respectively, the sets B_(x) := \ and B_(x) := \. 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 B is a vector subspace of the TVS X then this list may be extended to include:
  1. B is contained in the closure of \. * In other words, a vector subspace of X is bounded if and only if it is a subset of (the vector space) \operatorname_X \. * Recall that X 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 X. 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 X is called the or the () A or of X is a set \mathcal of bounded subsets of X such that every bounded subset of X is a subset of some B \in \mathcal. The set of all bounded subsets of X trivially forms a fundamental system of bounded sets of X.


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 ...
. 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 ...
X having a bounded subset B and also a dense vector subspace M such that B is contained in the closure (in X) of any bounded subset of M.


Stability properties


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 B_1, B_2, B_3, \ldots 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 t_1, t_2, t_3, \ldots of positive real numbers such that t_1 B_1, \, t_2 B_2, \, t_3 B_3, \ldots 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 ...
\mathcal 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 ...
Y is said to be in Y, if there exists some bounded subset D of Y such that B \subseteq D \quad \text B \in \mathcal, which happens if and only if its union \cup \mathcal ~:=~ \bigcup_ B is a bounded subset of Y. In the case of a normed (or seminormed) space, a family \mathcal is uniformly bounded if and only if its union \cup \mathcal is ''norm bounded'', meaning that there exists some real M \geq 0 such that \, b\, \leq M for every b \in \cup \mathcal, or equivalently, if and only if \sup_ \, b\, < \infty. A set H of maps from X to Y is said to be C \subseteq X if the family H(C) := \ is uniformly bounded in Y, which by definition means that there exists some bounded subset D of Y such that h(C) \subseteq D \text h \in H, or equivalently, if and only if \cup H(C) := \bigcup_ h(C) is a bounded subset of Y. A set H of linear maps between two normed (or seminormed) spaces X and Y is uniformly bounded on some (or equivalently, every) open ball (and/or non-degenerate closed ball) in X 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 \sup_ \, h\, < \infty. Assume H is equicontinuous and let W be a neighborhood of the origin in Y. Since H is equicontinuous, there exists a neighborhood U of the origin in X such that h(U) \subseteq W for every h \in H. Because C is bounded in X, there exists some real r > 0 such that if t \geq r then C \subseteq t U. So for every h \in H and every t \geq r, h(C) \subseteq h(t U) = t h(U) \subseteq t W, which implies that \bigcup_ h(C) \subseteq t W. Thus \bigcup_ h(C) is bounded in Y.
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 W be a balanced neighborhood of the origin in Y and let V be a closed balanced neighborhood of the origin in Y such that V + V \subseteq W. Define E ~:=~ \bigcap_ h^(V), which is a closed subset of X (since V is closed while every h : X \to Y is continuous) that satisfies h(E) \subseteq V for every h \in H. Note that for every non-zero scalar n \neq 0, the set n E is closed in X (since scalar multiplication by n \neq 0 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 C \cap n E is closed in C. It will now be shown that C \subseteq \bigcup_ n E, from which C = \bigcup_ (C \cap n E) follows. If c \in C then H(c) being bounded guarantees the existence of some positive integer n = n_c \in \N such that H(c) \subseteq n_c V, where the linearity of every h \in H now implies \tfrac c \in h^(V); thus \tfrac c \in \bigcap_ h^(V) = E and hence C \subseteq \bigcup_ n E, as desired. Thus C = (C \cap 1 E) \cup (C \cap 2 E) \cup (C \cap 3 E) \cup \cdots expresses C as a countable union of closed (in C) sets. Since C 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 n \in \N such that C \cap n E has non-empty interior in C. Let k \in \operatorname_C (C \cap n E) be any point belonging to this open subset of C. Let U be any balanced open neighborhood of the origin in X such that C \cap (k + U) ~\subseteq~ \operatorname_C (C \cap n E). The sets \ form an increasing (meaning p \leq q implies k + p U \subseteq k + q U) cover of the compact space C, so there exists some p > 1 such that C \subseteq k + p U (and thus \tfrac(C - k) \subseteq U). It will be shown that h(C) \subseteq p n W for every h \in H, thus demonstrating that \ is uniformly bounded in Y and completing the proof. So fix h \in H and c \in C. Let z ~:=~ \tfrac k + \tfrac c. The convexity of C guarantees z \in C and moreover, z \in k + U since z - k = \tfrac k + \tfrac c = \tfrac (c - k) \in \tfrac(C - k) \subseteq U. Thus z \in C \cap (k + U), which is a subset of \operatorname_C (C \cap n E). Since n V is balanced and , 1 - p, = p - 1 < p, we have (1 - p) n V \subseteq p n V, which combined with h(E) \subseteq V gives p n h(E) + (1 - p) n h(E) ~\subseteq~ p n V + (1 - p) n V ~\subseteq~ p n V + p n V ~\subseteq~ p n (V + V) ~\subseteq~ p n W. Finally, c = p z + (1 - p) k and k, z \in n E imply h(c) ~=~ p h(z) + (1 - p) h(k) ~\in~ p n h(E) + (1 - p) n h(E) ~\subseteq~ p n W, 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 X is also a bounded subset, it follows that if H \subseteq L(X, Y) 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 X and Y (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 ...
H(x) := \ of every x \in X is a bounded subset of Y.


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 A of a topological module M 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 ...
R is bounded if for any neighborhood N of 0_M there exists a neighborhood w of 0_R such that w A \subseteq B.


See also

* * * * * * * * *


References

Notes


Bibliography

* * * * * * * * * * * * * * * {{Topological vector spaces Topological vector spaces