Editing Equicontinuity (section)

==Equicontinuous linear maps==
{{anchor|Equicontinuous linear operators}}

Because every [[topological vector space]] (TVS) is a topological group, the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.

===Characterization of equicontinuous linear maps===

A family <math>H</math> of maps of the form <math>X \to Y</math> between two topological vector spaces is said to be {{em|equicontinuous at a point}} <math>x \in X</math> if for every neighborhood <math>V</math> of the origin in <math>Y</math> there exists some neighborhood <math>U</math> of the origin in <math>X</math> such that <math>h(x + U) \subseteq h(x) + V</math> for all <math>h \in H.</math> 

If <math>H</math> is a family of maps and <math>U</math> is a set then let <math>H(U) := \bigcup_{h \in H} h(U).</math> With notation, if <math>U</math> and <math>V</math> are sets then <math>h(U) \subseteq V</math> for all <math>h \in H</math> if and only if <math>H(U) \subseteq V.</math> 

Let <math>X</math> and <math>Y</math> be [[topological vector space]]s (TVSs) and <math>H</math> be a family of linear operators from <math>X</math> into <math>Y.</math> 
Then the following are equivalent: 
<ol>
<li><math>H</math> is equicontinuous;</li>
<li><math>H</math> is equicontinuous at every point of <math>X.</math> </li>
<li><math>H</math> is equicontinuous at some point of <math>X.</math> </li>
<li><math>H</math> is equicontinuous at the origin.
* that is, for every neighborhood <math>V</math> of the origin in <math>Y,</math> there exists a neighborhood <math>U</math> of the origin in <math>X</math> such that <math>H(U) \subseteq V</math> (or equivalently, <math>h(U) \subseteq V</math>for every <math>h \in H</math>).</li>{{sfn|Rudin|1991|p=44 Theorem 2.4}}
<li>for every neighborhood <math>V</math> of the origin in <math>Y,</math> <math>\bigcap_{h \in H} h^{-1}(V)</math> is a neighborhood of the origin in <math>X.</math> </li>
<li>the closure of <math>H</math> in <math>L_{\sigma}(X; Y)</math> is equicontinuous. 
* <math>L_{\sigma}(X; Y)</math> denotes <math>L(X; Y)</math>endowed with the topology of point-wise convergence.</li>
<li>the [[Balanced set|balanced hull]] of <math>H</math> is equicontinuous.</li>
</ol>

while if <math>Y</math> is [[locally convex]] then this list may be extended to include:
<ol start=8>
<li>the [[convex hull]] of <math>H</math> is equicontinuous.{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
<li>the [[Absolutely convex set|convex balanced hull]] of <math>H</math> is equicontinuous.{{sfn|Trèves|2006|pp=335-345}}{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
</ol>

while if <math>X</math> and <math>Y</math> are [[locally convex]] then this list may be extended to include: 
<ol start=10>
<li>for every continuous [[seminorm]] <math>q</math> on <math>Y,</math> there exists a continuous seminorm <math>p</math> on <math>X</math> such that <math>q \circ h \leq p</math> for all <math>h \in H.</math> {{sfn|Narici|Beckenstein|2011|pp=225-273}}
* Here, <math>q \circ h \leq p</math> means that <math>q(h(x)) \leq p(x)</math> for all <math>x \in X.</math></li>
</ol>

while if <math>X</math> is [[Barreled space|barreled]] and <math>Y</math> is locally convex then this list may be extended to include:
<ol start=11>
<li><math>H</math> is bounded in <math>L_{\sigma}(X; Y)</math>;{{sfn|Trèves|2006|pp=346-350}}</li>
<li><math>H</math> is bounded in <math>L_b(X; Y).</math> {{sfn|Trèves|2006|pp=346-350}}
* <math>L_b(X; Y)</math> denotes <math>L(X; Y)</math>endowed with the topology of bounded convergence (that is, uniform convergence on bounded subsets of <math>X.</math></li>
</ol>

while if <math>X</math> and <math>Y</math> are [[Banach space]]s then this list may be extended to include: 
<ol start=13>
<li><math>\sup \{\|T\| : T \in H\} < \infty</math> (that is, <math>H</math> is uniformly bounded in the [[operator norm]]).</li>
</ol>

====Characterization of equicontinuous linear functionals====
{{anchor|Equicontinuous linear functionals}}

Let <math>X</math> be a [[topological vector space]] (TVS) over the field <math>\mathbb{F}</math> with [[continuous dual space]] <math>X^{\prime}.</math> 
A family <math>H</math> of linear functionals on <math>X</math> is said to be {{em|equicontinuous at a point}} <math>x \in X</math> if for every neighborhood <math>V</math> of the origin in <math>\mathbb{F}</math> there exists some neighborhood <math>U</math> of the origin in <math>X</math> such that <math>h(x + U) \subseteq h(x) + V</math> for all <math>h \in H.</math> 

For any subset <math>H \subseteq X^{\prime},</math> the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
<ol>
<li><math>H</math> is equicontinuous.</li>
<li><math>H</math> is equicontinuous at the origin.</li>
<li><math>H</math> is equicontinuous at some point of <math>X.</math> </li>
<li><math>H</math> is contained in the [[Polar set|polar]] of some neighborhood of the origin in <math>X</math>{{sfn|Trèves|2006|pp=335-345}}</li>
<li>the [[Polar set|(pre)polar]] of <math>H</math> is a neighborhood of the origin in <math>X.</math> </li>
<li>the [[Weak-* topology|weak* closure]] of <math>H</math> in <math>X^{\prime}</math> is equicontinuous.</li>
<li>the [[Balanced set|balanced hull]] of <math>H</math> is equicontinuous.</li>
<li>the [[convex hull]] of <math>H</math> is equicontinuous.</li>
<li>the [[Absolutely convex set|convex balanced hull]] of <math>H</math> is equicontinuous.{{sfn|Trèves|2006|pp=335-345}}</li>
</ol>

while if <math>X</math> is [[Normed space|normed]] then this list may be extended to include:
<ol start=10>
<li><math>H</math> is a strongly bounded subset of <math>X^{\prime}.</math>{{sfn|Trèves|2006|pp=335-345}}</li>
</ol>

while if <math>X</math> is a [[barreled space]] then this list may be extended to include:
<ol start=11>
<li><math>H</math> is [[relatively compact]] in the [[weak* topology]] on <math>X^{\prime}.</math>{{sfn|Trèves|2006|pp=346-350}}</li>
<li><math>H</math> is [[weak* topology|weak* bounded]] (that is, <math>H</math> is <math>\sigma\left(X^{\prime}, X\right)-</math>bounded in <math>X^{\prime}</math>).{{sfn|Trèves|2006|pp=346-350}}</li>
<li><math>H</math> is bounded in the topology of bounded convergence (that is, <math>H</math> is <math>b\left(X^{\prime}, X\right)-</math>bounded in <math>X^{\prime}</math>).{{sfn|Trèves|2006|pp=346-350}}</li>
</ol>

===Properties of equicontinuous linear maps===

The [[uniform boundedness principle]] (also known as the Banach–Steinhaus theorem) states that a set <math>H</math> of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is, <math>\sup_{h \in H} \|h(x)\| < \infty</math> for each <math>x \in X.</math> The result can be generalized to a case when <math>Y</math> is locally convex and <math>X</math> is a [[barreled space]].{{sfn|Schaefer|1966|loc= Theorem 4.2}}

====Properties of equicontinuous linear functionals====

[[Alaoglu's theorem]] implies that the weak-* closure of an equicontinuous subset of <math>X^{\prime}</math> is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.{{sfn|Schaefer|1966|loc= Corollary 4.3}}{{sfn|Narici|Beckenstein|2011|pp=225-273}}

If <math>X</math> is any locally convex TVS, then the family of all [[Barrelled space|barrel]]s in <math>X</math> and the family of all subsets of <math>X^{\prime}</math> that are convex, balanced, closed, and bounded in <math>X^{\prime}_{\sigma},</math> correspond to each other by polarity (with respect to <math>\left\langle X, X^{\#} \right\rangle</math>).{{sfn|Schaefer|Wolff|1999|pp=123–128}} 
It follows that a locally convex TVS <math>X</math> is barreled if and only if every bounded subset of <math>X^{\prime}_{\sigma}</math> is equicontinuous.{{sfn|Schaefer|Wolff|1999|pp=123–128}}

{{Math theorem|name=Theorem|math_statement=
Suppose that <math>X</math> is a [[Separable space|separable]] TVS. Then every closed equicontinuous subset of <math>X^{\prime}_{\sigma}</math> is a compact metrizable space (under the subspace topology). If in addition <math>X</math> is metrizable then <math>X^{\prime}_{\sigma}</math> is separable.{{sfn|Schaefer|Wolff|1999|pp=123–128}} 
}}