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Equicontinuity
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====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>
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