View source for Bounded set (topological vector space)

{{Short description|Generalization of boundedness}}
{{for|bounded sets in general|bounded set}}

In [[functional analysis]] and related areas of [[mathematics]], a [[Set (mathematics)|set]] in a [[topological vector space]] is called '''bounded''' or '''von Neumann bounded''', if every [[Neighbourhood (mathematics)|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 topological vector space|locally convex]] [[Polar topology|polar topologies]] on the [[vector space]]s in a [[dual pair]], as the [[polar set]] of a bounded set is an [[Absolutely convex set|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 [[topological field]] <math>\mathbb{K}.</math> 

A subset <math>B</math> of <math>X</math> is called '''{{em|[[John von Neumann|von Neumann]] bounded}}''' or just '''{{em|bounded}}''' in <math>X</math> if any of the following equivalent conditions are satisfied:
<ol>
<li>{{em|Definition}}: 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><ref group=note>For any set <math>A</math> and [[Scalar (mathematics)|scalar]] <math>s,</math> the notation <math>s A</math> denotes the set <math>s A := \{s a : a \in A\}.</math></ref> for all scalars <math>s</math> satisfying <math>|s| \geq r.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
* This was the definition introduced by [[John von Neumann]] in 1935.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li><math>B</math> is [[Absorbing set|absorbed]] by every [[Neighborhood (mathematics)|neighborhood]] of the origin.{{sfn|Schaefer|1970|p=25}}</li>
<li>For every neighborhood <math>V</math> of the origin there exists a scalar <math>s</math> such that <math>B \subseteq s V.</math></li>
<li>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>{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>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>{{sfn|Rudin|1991|p=8}}</li>
<li>Any one of statements (1) through (5) above but with the word "neighborhood" replaced by any of the following: "[[Balanced set|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 set|balanced]] neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=156-175}}
* If <math>X</math> is [[Locally convex topological vector space|locally convex]] then the adjective "convex" may be also be added to any of these 5 replacements.</li>
<li>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>{{sfn|Narici|Beckenstein|2011|pp=156-175}}
* 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.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>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>{{sfn|Wilansky|2013|p=47}}</li>
<li>Every [[Countable set|countable]] subset of <math>B</math> is bounded (according to any defining condition other than this one).{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
</ol>

If <math>\mathcal{B}</math> is a [[neighborhood basis]] for <math>X</math> at the origin then this list may be extended to include:
<ol start=10>
<li>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></li>
</ol>

If <math>X</math> is a [[Locally convex topological vector space|locally convex]] space whose topology is defined by a family <math>\mathcal{P}</math> of continuous [[seminorm]]s, then this list may be extended to include:
<ol start=11>
<li><math>p(B)</math> is bounded for all <math>p \in \mathcal{P}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>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).{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>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></li>
<li>B is weakly bounded, i.e. every continuous linear functional is bounded on B<ref>{{Cite book |last=Narici Beckenstein |title=Topological Vector Spaces |year=2011 |isbn=978-1-58488-866-6 |edition=2nd |pages=253, Theorem 8.8.7}}</ref></li>
</ol>

If <math>X</math> is a [[normed space]] with [[Norm (mathematics)|norm]] <math>\|\cdot\|</math> (or more generally, if it is a [[seminormed space]] and <math>\|\cdot\|</math> is merely a [[seminorm]]),<ref group=note>This means that the topology on <math>X</math> is equal to the topology induced on it by <math>\|\cdot\|.</math> Note that every [[normed space]] is a seminormed space and every [[Norm (mathematics)|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.</ref> then this list may be extended to include:
<ol start=14>
<li><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>{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li><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></li>
<li><math>B</math> is a subset of some (open or closed) ball.<ref group=note name="openclosedballdefinition">If <math>(X, \|\cdot\|)</math> is a [[normed space]] or a [[seminormed space]], then the open and closed balls of radius <math>r > 0</math> (where <math>r \neq \infty</math> is a real number) centered at a point <math>x \in X</math> are, respectively, the sets <math display=inline>B_{< r}(x) := \{z \in X : \|z - x\| < r\}</math> and <math display=inline>B_{\leq r}(x) := \{z \in X : \|z - x\| \leq r\}.</math> Any such set is called a (non-degenerate) {{em|ball}}.</ref>
* This ball need not be centered at the origin, but its radius must (as usual) be positive and finite.</li>
</ol>

If <math>B</math> is a vector subspace of the TVS <math>X</math> then this list may be extended to include:
<ol start=17>
<li><math>B</math> is contained in the closure of <math>\{0\}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}
* 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></li>
</ol>

A subset that is not bounded is called {{em|unbounded}}.

===Bornology and fundamental systems of bounded sets===

The collection of all bounded sets on a topological vector space <math>X</math> is called the {{em|von Neumann bornology}} or the ({{em|canonical}}) {{em|[[Bornological space#Bornology of a topological vector space|bornology]] of <math>X.</math>}}

A {{em|base}} or {{em|fundamental system of bounded sets}} 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>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
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 topological vector space|locally convex]] TVS, the set of closed and bounded [[Absolutely convex set|disks]] are a base of bounded set.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

==Examples and sufficient conditions==

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

<ul>
<li>Finite sets are bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>Every [[totally bounded]] subset of a TVS is bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>Every [[relatively compact set]] in a topological vector space is bounded. If the space is equipped with the [[Weak topology (polar topology)|weak topology]] the converse is also true.</li>
<li>The set of points of a [[Cauchy sequence]] is bounded, the set of points of a [[Cauchy net|Cauchy]] [[Net (mathematics)|net]] need not be bounded.</li>
<li>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></li>
</ul>

'''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 {{em|not}} contained in the closure (in <math>X</math>) of any bounded subset of <math>M.</math>{{sfn|Wilansky|2013|p=57}}

===Stability properties===

<ul>
<li>In any TVS, finite [[Union (set theory)|unions]], finite [[Minkowski sum]]s, scalar multiples, translations, subsets, [[Closure (topology)|closures]], [[Interior (topology)|interiors]], and [[Balanced set|balanced hull]]s of bounded sets are again bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>In any [[Locally convex topological vector space|locally convex TVS]], the [[Convex set|convex hull]] (also called the [[convex envelope]]) of a bounded set is again bounded.{{sfn|Narici|Beckenstein|2011|p=162}} 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.{{sfn|Narici|Beckenstein|2011|p=162}}</li>
<li>The image of a bounded set under a [[continuous linear map]] is a bounded subset of the codomain.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>A subset of an arbitrary [[Product space|(Cartesian) product]] of TVSs is bounded if and only if its image under every coordinate projections is bounded.</li>
<li>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>{{sfn|Narici|Beckenstein|2011|pp=156-175}}
* 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></li>
</ul>

==Properties==

{{See also|Topological vector space#Properties}}

A [[locally convex topological vector space]] has a bounded neighborhood of zero [[if and only if]] its topology can be defined by a {{em|single}} [[seminorm]].

The [[Polar set|polar]] of a bounded set is an [[Absolutely convex set|absolutely convex]] and [[absorbing set]].

{{Math theorem|name={{visible anchor|Mackey's countability condition}}{{sfn|Narici|Beckenstein|2011|p=174}}|note=|math_statement=
If <math>B_1, B_2, B_3, \ldots</math> is a countable sequence of bounded subsets of a [[Metrizable topological vector space|metrizable]] [[locally convex topological vector space]] <math>X,</math> then there exists a bounded subset <math>B</math> of <math>X</math> and a sequence <math>r_1, r_2, r_3, \ldots</math> of positive real numbers such that <math>B_i \subseteq r_i B</math> for all <math>i \in \N</math> (or equivalently, such that <math>\tfrac{1}{r_1} B_1 \cup \tfrac{1}{r_2} B_2 \cup \tfrac{1}{r_3} B_3 \cup \cdots \subseteq B</math>).
}}

Using the definition of [[#uniformly bounded sets|uniformly bounded sets]] given below, [[#Mackey's countability condition|Mackey's countability condition]] can be restated as: If <math>B_1, B_2, B_3, \ldots</math> are bounded subsets of a [[Metrizable topological vector space|metrizable]] [[Locally convex topological vector space|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 sets|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===

{{See also|Uniform boundedness principle}}

A [[family of sets]] <math>\mathcal{B}</math> of subsets of a [[topological vector space]] <math>Y</math> is said to be {{em|{{visible anchor|uniformly bounded|uniformly bounded sets|uniformly bounded family}}}} 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 space|normed]] (or [[Seminormed space|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 {{em|{{visible anchor|uniformly bounded maps on a set|text=uniformly bounded on a given set}}}} <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 norm]]s are uniformly bounded; that is, if and only if <math display=inline>\sup_{h \in H} \|h\| < \infty.</math> 

{{math theorem | name = Proposition{{sfn|Rudin|1991|pp=42−47}} | math_statement=
Let <math>H \subseteq L(X, Y)</math> be a set of continuous linear operators between two [[topological vector space]]s <math>X</math> and <math>Y</math> and let <math>C \subseteq X</math> be any bounded subset of <math>X.</math> 
Then <math>H</math> is uniformly bounded on <math>C</math> (that is, the [[family of sets|family]] <math>\{h(C) : h \in H\}</math> is uniformly bounded in <math>Y</math>) if any of the following conditions are satisfied:
# <math>H</math> is [[Equicontinuity|equicontinuous]].
# <math>C</math> is a [[Convex set|convex]] [[Compact space|compact]] Hausdorff [[Subspace (topology)|subspace]] of <math>X</math> and for every <math>c \in C,</math> the [[Orbit (group theory)|orbit]] <math>H(c) := \{h(c) : h \in H\}</math> is a bounded subset of <math>Y.</math>
}}

{{collapse top|title=Proof of part (1){{sfn|Rudin|1991|pp=42−47}}|left=true|collapse=no}} 
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.]]
{{collapse bottom}} 

{{collapse top|title=Proof of part (2){{sfn|Rudin|1991|pp=46−47}}|left=true}} 
Let <math>W</math> be a [[Balanced set|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.]]
{{collapse bottom}}

Since every [[Singleton set|singleton subset]] of <math>X</math> is also a bounded subset, it follows that if <math>H \subseteq L(X, Y)</math> is an [[Equicontinuity|equicontinuous]] set of [[continuous linear operator]]s between two [[topological vector space]]s <math>X</math> and <math>Y</math> (not necessarily [[Hausdorff space|Hausdorff]] or locally convex), then the [[Orbit (group theory)|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 module]]s. 
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>

==See also==

* {{annotated link|Bornological space}}
* {{annotated link|Bornivorous set}}
* {{annotated link|Bounded function}}
* {{annotated link|Bounded operator}}
* {{annotated link|Bounding point}}
* {{annotated link|Compact space}}
* {{annotated link|Kolmogorov's normability criterion}}
* {{annotated link|Local boundedness}}
* {{annotated link|Totally bounded space}}

==References==

{{reflist}}

'''Notes'''

{{reflist|group=note}}

==Bibliography==

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* {{Berberian Lectures in Functional Analysis and Operator Theory}} <!-- {{sfn|Berberian|2014|p=}} -->
* {{Bourbaki Topological Vector Spaces}} <!-- {{sfn|Bourbaki|1987|p=}} -->
* {{Conway A Course in Functional Analysis}} <!-- {{sfn|Conway|1990|p=}} -->
* {{Edwards Functional Analysis Theory and Applications}} <!-- {{sfn|Edwards|1995|p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn|Grothendieck|1973|p=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1969|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} 
* {{cite book|last=Robertson|first=A.P.|author2=W.J. Robertson|title=Topological vector spaces|series=Cambridge Tracts in Mathematics|volume=53|year=1964|publisher=[[Cambridge University Press]]|pages=44–46}}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Robertson Topological Vector Spaces}} <!-- {{sfn|Robertson|Robertson|1980|p=}} -->
* {{cite book|first=H.H.|last=Schaefer|title=Topological Vector Spaces|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics|GTM]]|volume=3|date=1970|isbn=0-387-05380-8|pages=25–26}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional analysis}}
{{Boundedness and bornology}}
{{Topological vector spaces}}

[[Category:Topological vector spaces]]