Editing Reflexive space (section)

=== Characterizations of reflexive spaces ===

If <math>X</math> is a Hausdorff locally convex space then the following are equivalent:
#<math>X</math> is reflexive;
#<math>X</math> is [[Semireflexive space|semireflexive]] and [[Infrabarreled space|infrabarreled]];{{sfn|Narici|Beckenstein|2011|pp=488-491}}
#<math>X</math> is [[Semireflexive space|semireflexive]] and [[barreled space|barreled]];
#<math>X</math> is [[barreled space|barreled]] and the weak topology on <math>X</math> had the Heine-Borel property (that is, for the weak topology <math>\sigma\left(X, X^{\prime}\right),</math> every closed and bounded subset of <math>X_{\sigma}</math> is weakly compact).{{sfn|Trèves|2006|pp=372-374}}
#<math>X</math> is [[Semireflexive space|semireflexive]] and [[Quasibarrelled space|quasibarrelled]].{{sfn|Khaleelulla|1982|pp=32-63}}

If <math>X</math> is a normed space then the following are equivalent:
#<math>X</math> is reflexive;
#The closed unit ball is compact when <math>X</math> has the weak topology <math>\sigma\left(X, X^{\prime}\right).</math>{{sfn|Trèves|2006|p=376}}
#<math>X</math> is a Banach space and <math>X^{\prime}_b</math> is reflexive.{{sfn|Trèves|2006|p=377}}
#Every sequence <math>\left(C_n\right)_{n=1}^{\infty},</math> with <math>C_{n+1} \subseteq C_n</math> for all <math>n</math> of nonempty closed bounded convex subsets of <math>X</math> has nonempty intersection.{{sfn|Bernardes| 2012|p=}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=212}}|math_statement=
A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be [[Hahn–Banach theorem|strictly separated by a hyperplane]].
}}

{{Math theorem|name=[[James's theorem|James' theorem]]|math_statement=
A [[Banach space]] <math>B</math> is reflexive if and only if every [[Continuous function|continuous]] [[linear functional]] on <math>B</math> attains its [[supremum]] on the closed [[unit ball]] in <math>B.</math>
}}