Английская Википедия:Bounded set (topological vector space)

In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector 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 polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.

Definition

Suppose <math>X</math> is a topological vector space (TVS) over a field <math>\mathbb{K}.</math>

A subset <math>B</math> of <math>X</math> is called Шаблон:Em or just Шаблон:Em in <math>X</math> if any of the following equivalent conditions are satisfied:

Шаблон:Em: For every neighborhood <math>V</math> of the origin there exists a real <math>r > 0</math> such that <math>B \subseteq s V</math>^{[note 1]} for all scalars <math>s</math> satisfying <math>|s| \geq r.</math>Шаблон:Sfn
- This was the definition introduced by John von Neumann in 1935.Шаблон:Sfn
<math>B</math> is absorbed by every neighborhood of the origin.Шаблон:Sfn
For every neighborhood <math>V</math> of the origin there exists a scalar <math>s</math> such that <math>B \subseteq s V.</math>
For every neighborhood <math>V</math> of the origin there exists a real <math>r > 0</math> such that <math>s B \subseteq V</math> for all scalars <math>s</math> satisfying <math>|s| \leq r.</math>Шаблон:Sfn
For every neighborhood <math>V</math> of the origin there exists a real <math>r > 0</math> such that <math>t B \subseteq V</math> for all real <math>0 < t \leq r.</math>Шаблон:Sfn
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: <math>B</math> is bounded if and only if <math>B</math> is absorbed by every balanced neighborhood of the origin.Шаблон:Sfn
- If <math>X</math> is locally convex then the adjective "convex" may be also be added to any of these 5 replacements.
For every sequence of scalars <math>s_1, s_2, s_3, \ldots</math> that converges to <math>0</math> and every sequence <math>b_1, b_2, b_3, \ldots</math> in <math>B,</math> the sequence <math>s_1 b_1, s_2 b_2, s_3 b_3, \ldots</math> converges to <math>0</math> in <math>X.</math>Шаблон:Sfn
- This was the definition of "bounded" that Andrey Kolmogorov used in 1934, which is the same as the definition introduced by Stanisław Mazur and Władysław Orlicz 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.Шаблон:Sfn
For every sequence <math>b_1, b_2, b_3, \ldots</math> in <math>B,</math> the sequence <math display=inline>\left(\tfrac{1}{i} b_i\right)_{i=1}^{\infty}</math> converges to <math>0</math> in <math>X.</math>Шаблон:Sfn
Every countable subset of <math>B</math> is bounded (according to any defining condition other than this one).Шаблон:Sfn

If <math>\mathcal{B}</math> is a neighborhood basis for <math>X</math> 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 <math>\mathcal{B}.</math>
- e.g. Statement (3) may become: For every <math>V \in \mathcal{B}</math> there exists a scalar <math>s</math> such that <math>B \subseteq s V.</math>

If <math>X</math> is a locally convex space whose topology is defined by a family <math>\mathcal{P}</math> of continuous seminorms, then this list may be extended to include:

<math>p(B)</math> is bounded for all <math>p \in \mathcal{P}.</math>Шаблон:Sfn
There exists a sequence of non-zero scalars <math>s_1, s_2, s_3, \ldots</math> such that for every sequence <math>b_1, b_2, b_3, \ldots</math> in <math>B,</math> the sequence <math>b_1 s_1, b_2 s_2, b_3 s_3, \ldots</math> is bounded in <math>X</math> (according to any defining condition other than this one).Шаблон:Sfn
For all <math>p \in \mathcal{P},</math> <math>B</math> is bounded (according to any defining condition other than this one) in the semi normed space <math>(X, p).</math>
B is weakly bounded, i.e. every continuous linear functional is bounded on B^[1]

If <math>X</math> is a normed space with norm <math>\|\cdot\|</math> (or more generally, if it is a seminormed space and <math>\|\cdot\|</math> is merely a seminorm),^{[note 2]} then this list may be extended to include:

<math>B</math> is a norm bounded subset of <math>(X, \|\cdot\|).</math> By definition, this means that there exists a real number <math>r > 0</math> such that <math>\|b\| \leq r</math> for all <math>b \in B.</math>Шаблон:Sfn
<math>\sup_{b \in B} \|b\| < \infty.</math>
- Thus, if <math>L : (X, \|\cdot\|) \to (Y, \|\cdot\|)</math> is a linear map between two normed (or seminormed) spaces and if <math>B</math> is the closed (alternatively, open) unit ball in <math>(X, \|\cdot\|)</math> centered at the origin, then <math>L</math> is a bounded linear operator (which recall means that its operator norm <math>\|L\| := \sup_{b \in B} \|L(b)\| < \infty</math> is finite) if and only if the image <math>L(B)</math> of this ball under <math>L</math> is a norm bounded subset of <math>(Y, \|\cdot\|).</math>
<math>B</math> is a subset of some (open or closed) ball.^{[note 3]}
- This ball need not be centered at the origin, but its radius must (as usual) be positive and finite.

If <math>B</math> is a vector subspace of the TVS <math>X</math> then this list may be extended to include:

<math>B</math> is contained in the closure of <math>\{0\}.</math>Шаблон:Sfn
- In other words, a vector subspace of <math>X</math> is bounded if and only if it is a subset of (the vector space) <math>\operatorname{cl}_X \{0\}.</math>
- Recall that <math>X</math> is a Hausdorff space if and only if <math>\{0\}</math> is closed in <math>X.</math> So the only bounded vector subspace of a Hausdorff TVS is <math>\{0\}.</math>

A subset that is not bounded is called Шаблон:Em.

Bornology and fundamental systems of bounded sets

The collection of all bounded sets on a topological vector space <math>X</math> is called the Шаблон:Em or the (Шаблон:Em) Шаблон:Em

A Шаблон:Em or Шаблон:Em of <math>X</math> is a set <math>\mathcal{B}</math> of bounded subsets of <math>X</math> such that every bounded subset of <math>X</math> is a subset of some <math>B \in \mathcal{B}.</math>Шаблон:Sfn The set of all bounded subsets of <math>X</math> trivially forms a fundamental system of bounded sets of <math>X.</math>

Examples

In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.Шаблон:Sfn

Examples and sufficient conditions

Unless indicated otherwise, a topological vector space (TVS) need not be Hausdorff nor locally convex.

Finite sets are bounded.Шаблон:Sfn
Every totally bounded subset of a TVS is bounded.Шаблон:Sfn
Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topology the converse is also true.
The set of points of a Cauchy sequence 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 <math>\{0\}</math>) is always a bounded closed vector subspace. This set <math>\operatorname{cl}_X \{0\}</math> is the unique largest (with respect to set inclusion <math>\,\subseteq\,</math>) bounded vector subspace of <math>X.</math> In particular, if <math>B \subseteq X</math> is a bounded subset of <math>X</math> then so is <math>B + \operatorname{cl}_X \{0\}.</math>

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 <math>\{0\}</math> is unbounded

There exists a Fréchet space <math>X</math> having a bounded subset <math>B</math> and also a dense vector subspace <math>M</math> such that <math>B</math> is Шаблон:Em contained in the closure (in <math>X</math>) of any bounded subset of <math>M.</math>Шаблон:Sfn

Stability properties

In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded.Шаблон:Sfn
In any locally convex TVS, the convex hull (also called the convex envelope) of a bounded set is again bounded.Шаблон:Sfn However, this may be false if the space is not locally convex, as the (non-locally convex) Lp space <math>L^p</math> spaces for <math>0 < p < 1</math> have no nontrivial open convex subsets.Шаблон:Sfn
The image of a bounded set under a continuous linear map is a bounded subset of the codomain.Шаблон:Sfn
A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded.
If <math>S \subseteq X \subseteq Y</math> and <math>X</math> is a topological vector subspace of <math>Y,</math> then <math>S</math> is bounded in <math>X</math> if and only if <math>S</math> is bounded in <math>Y.</math>Шаблон:Sfn
- In other words, a subset <math>S \subseteq X</math> is bounded in <math>X</math> if and only if it is bounded in every (or equivalently, in some) topological vector superspace of <math>X.</math>

Properties

Шаблон:See also

A locally convex topological vector space has a bounded neighborhood of zero if and only if its topology can be defined by a Шаблон:Em seminorm.

The polar of a bounded set is an absolutely convex and absorbing set.

Шаблон:Math theorem

Using the definition of uniformly bounded sets given below, Mackey's countability condition can be restated as: If <math>B_1, B_2, B_3, \ldots</math> are bounded subsets of a metrizable locally convex space then there exists a sequence <math>t_1, t_2, t_3, \ldots</math> of positive real numbers such that <math>t_1 B_1, \, t_2 B_2, \, t_3 B_3, \ldots</math> 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

Шаблон:See also

A family of sets <math>\mathcal{B}</math> of subsets of a topological vector space <math>Y</math> is said to be Шаблон:Em in <math>Y,</math> if there exists some bounded subset <math>D</math> of <math>Y</math> such that <math display=block>B \subseteq D \quad \text{ for every } B \in \mathcal{B},</math> which happens if and only if its union <math display=block>\cup \mathcal{B} ~:=~ \bigcup_{B \in \mathcal{B}} B</math> is a bounded subset of <math>Y.</math> In the case of a normed (or seminormed) space, a family <math>\mathcal{B}</math> is uniformly bounded if and only if its union <math>\cup \mathcal{B}</math> is norm bounded, meaning that there exists some real <math>M \geq 0</math> such that <math display=inline>\|b\| \leq M</math> for every <math>b \in \cup \mathcal{B},</math> or equivalently, if and only if <math display=inline>\sup_{\stackrel{b \in B}{B \in \mathcal{B}}} \|b\| < \infty.</math>

A set <math>H</math> of maps from <math>X</math> to <math>Y</math> is said to be Шаблон:Em <math>C \subseteq X</math> if the family <math>H(C) := \{h(C) : h \in H\}</math> is uniformly bounded in <math>Y,</math> which by definition means that there exists some bounded subset <math>D</math> of <math>Y</math> such that <math>h(C) \subseteq D \text{ for all } h \in H,</math> or equivalently, if and only if <math display=inline>\cup H(C) := \bigcup_{h \in H} h(C)</math> is a bounded subset of <math>Y.</math> A set <math>H</math> of linear maps between two normed (or seminormed) spaces <math>X</math> and <math>Y</math> is uniformly bounded on some (or equivalently, every) open ball (and/or non-degenerate closed ball) in <math>X</math> if and only if their operator norms are uniformly bounded; that is, if and only if <math display=inline>\sup_{h \in H} \|h\| < \infty.</math>

Шаблон:Math theorem

Шаблон:Collapse top Assume <math>H</math> is equicontinuous and let <math>W</math> be a neighborhood of the origin in <math>Y.</math> Since <math>H</math> is equicontinuous, there exists a neighborhood <math>U</math> of the origin in <math>X</math> such that <math>h(U) \subseteq W</math> for every <math>h \in H.</math> Because <math>C</math> is bounded in <math>X,</math> there exists some real <math>r > 0</math> such that if <math>t \geq r</math> then <math>C \subseteq t U.</math> So for every <math>h \in H</math> and every <math>t \geq r,</math> <math>h(C) \subseteq h(t U) = t h(U) \subseteq t W,</math> which implies that <math display=inline>\bigcup_{h \in H} h(C) \subseteq t W.</math> Thus <math display=inline>\bigcup_{h \in H} h(C)</math> is bounded in <math>Y.</math> Q.E.D. Шаблон:Collapse bottom

Шаблон:Collapse top Let <math>W</math> be a balanced neighborhood of the origin in <math>Y</math> and let <math>V</math> be a closed balanced neighborhood of the origin in <math>Y</math> such that <math>V + V \subseteq W.</math> Define <math display=block>E ~:=~ \bigcap_{h \in H} h^{-1}(V),</math> which is a closed subset of <math>X</math> (since <math>V</math> is closed while every <math>h : X \to Y</math> is continuous) that satisfies <math>h(E) \subseteq V</math> for every <math>h \in H.</math> Note that for every non-zero scalar <math>n \neq 0,</math> the set <math>n E</math> is closed in <math>X</math> (since scalar multiplication by <math>n \neq 0</math> is a homeomorphism) and so every <math>C \cap n E</math> is closed in <math>C.</math>

It will now be shown that <math>C \subseteq \bigcup_{n \in \N} n E,</math> from which <math>C = \bigcup_{n \in \N} (C \cap n E)</math> follows. If <math>c \in C</math> then <math>H(c)</math> being bounded guarantees the existence of some positive integer <math>n = n_c \in \N</math> such that <math>H(c) \subseteq n_c V,</math> where the linearity of every <math>h \in H</math> now implies <math>\tfrac{1}{n_c} c \in h^{-1}(V);</math> thus <math>\tfrac{1}{n_c} c \in \bigcap_{h \in H} h^{-1}(V) = E</math> and hence <math>C \subseteq \bigcup_{n \in \N} n E,</math> as desired.

Thus <math display=inline>C = (C \cap 1 E) \cup (C \cap 2 E) \cup (C \cap 3 E) \cup \cdots</math> expresses <math>C</math> as a countable union of closed (in <math>C</math>) sets. Since <math>C</math> is a nonmeager subset of itself (as it is a Baire space by the Baire category theorem), this is only possible if there is some integer <math>n \in \N</math> such that <math>C \cap n E</math> has non-empty interior in <math>C.</math> Let <math>k \in \operatorname{Int}_C (C \cap n E)</math> be any point belonging to this open subset of <math>C.</math> Let <math>U</math> be any balanced open neighborhood of the origin in <math>X</math> such that <math display=block>C \cap (k + U) ~\subseteq~ \operatorname{Int}_C (C \cap n E).</math>

The sets <math>\{k + p U : p > 1\}</math> form an increasing (meaning <math>p \leq q</math> implies <math>k + p U \subseteq k + q U</math>) cover of the compact space <math>C,</math> so there exists some <math>p > 1</math> such that <math>C \subseteq k + p U</math> (and thus <math>\tfrac{1}{p}(C - k) \subseteq U</math>). It will be shown that <math>h(C) \subseteq p n W</math> for every <math>h \in H,</math> thus demonstrating that <math>\{h(C) : h \in H\}</math> is uniformly bounded in <math>Y</math> and completing the proof. So fix <math>h \in H</math> and <math>c \in C.</math> Let <math display=block>z ~:=~ \tfrac{p - 1}{p} k + \tfrac{1}{p} c.</math>

The convexity of <math>C</math> guarantees <math>z \in C</math> and moreover, <math>z \in k + U</math> since <math display=block>z - k = \tfrac{-1}{p} k + \tfrac{1}{p} c = \tfrac{1}{p} (c - k) \in \tfrac{1}{p}(C - k) \subseteq U.</math> Thus <math>z \in C \cap (k + U),</math> which is a subset of <math>\operatorname{Int}_C (C \cap n E).</math> Since <math>n V</math> is balanced and <math>|1 - p| = p - 1 < p,</math> we have <math>(1 - p) n V \subseteq p n V,</math> which combined with <math>h(E) \subseteq V</math> gives <math display=block>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.</math> Finally, <math>c = p z + (1 - p) k</math> and <math>k, z \in n E</math> imply <math display=block>h(c) ~=~ p h(z) + (1 - p) h(k) ~\in~ p n h(E) + (1 - p) n h(E) ~\subseteq~ p n W,</math> as desired. Q.E.D. Шаблон:Collapse bottom

Since every singleton subset of <math>X</math> is also a bounded subset, it follows that if <math>H \subseteq L(X, Y)</math> is an equicontinuous set of continuous linear operators between two topological vector spaces <math>X</math> and <math>Y</math> (not necessarily Hausdorff or locally convex), then the orbit <math display=inline>H(x) := \{h(x) : h \in H\}</math> of every <math>x \in X</math> is a bounded subset of <math>Y.</math>

Bounded subsets of topological modules

The definition of bounded sets can be generalized to topological modules. A subset <math>A</math> of a topological module <math>M</math> over a topological ring <math>R</math> is bounded if for any neighborhood <math>N</math> of <math>0_M</math> there exists a neighborhood <math>w</math> of <math>0_R</math> such that <math>w A \subseteq B.</math>

References

Шаблон:Reflist

Notes

Шаблон:Reflist

Bibliography

Шаблон:Functional analysis Шаблон:Boundedness and bornology Шаблон:Topological vector spaces

Ошибка цитирования Для существующих тегов <ref> группы «note» не найдено соответствующего тега <references group="note"/>

↑ Шаблон:Cite book

[2] Шаблон:Cite book

[note 1]

[1]

[note 2]

[note 3]

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Английская Википедия:Bounded set (topological vector space)

Содержание

Definition

Bornology and fundamental systems of bounded sets

Examples

Examples and sufficient conditions

Stability properties

Properties

Generalizations

Uniformly bounded sets

Bounded subsets of topological modules

See also

References

Bibliography

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