Editing Equicontinuity (section)

===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}} 
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