Editing Linear form (section)

=== Equicontinuity of families of linear functionals ===
Let {{mvar|X}} be a [[topological vector space]] (TVS) with [[continuous dual space]] <math>X'.</math>

For any subset {{math|''H''}} of <math>X',</math> the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
# {{math|''H''}} is [[Equicontinuity|equicontinuous]];
# {{math|''H''}} is contained in the [[Polar set|polar]] of some neighborhood of <math>0</math> in {{mvar|X}};
# the [[Polar set|(pre)polar]] of {{math|''H''}} is a neighborhood of <math>0</math> in {{mvar|X}};

If {{math|''H''}} is an equicontinuous subset of <math>X'</math> then the following sets are also equicontinuous: 
the [[Weak-* topology|weak-*]] closure, the [[Balanced set|balanced hull]], the [[convex hull]], and the [[Absolutely convex set|convex balanced hull]].{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
Moreover, [[Alaoglu's theorem]] implies that the weak-* closure of an equicontinuous subset of  <math>X'</math> is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact).{{sfn|Schaefer|Wolff|1999|loc=Corollary 4.3}}{{sfn|Narici|Beckenstein|2011|pp=225-273}}