Editing Reflexive space (section)

=== Properties ===

Since every finite-dimensional normed space is a reflexive [[Banach space]], only infinite-dimensional spaces can be non-reflexive.

If a Banach space <math>Y</math> is isomorphic to a reflexive Banach space <math>X</math> then <math>Y</math> is reflexive.<ref>Proposition 1.11.8 in {{harvtxt|Megginson|1998|p=99}}.</ref>

Every [[Closed set|closed]] [[linear subspace]] of a reflexive space is reflexive.  The continuous dual of a reflexive space is reflexive.  Every [[Quotient space (linear algebra)|quotient]] of a reflexive space by a closed subspace is reflexive.<ref>{{harvtxt|Megginson|1998|pp=104&ndash;105}}.</ref>

Let <math>X</math> be a Banach space. The following are equivalent.
<ol>
<li>The space <math>X</math> is reflexive.</li>
<li>The continuous dual of <math>X</math> is reflexive.<ref>Corollary 1.11.17, p.&nbsp;104 in {{harvtxt|Megginson|1998}}.</ref></li>
<li>The closed unit ball of <math>X</math> is [[compact space|compact]] in the [[weak topology]]. (This is known as Kakutani's Theorem.){{sfn|Conway|1985|loc=Theorem V.4.2, p.&nbsp;135}}</li>
<li>Every bounded sequence in <math>X</math> has a weakly convergent subsequence.<ref>Since weak compactness and weak sequential compactness coincide by the [[Eberlein–Šmulian theorem]].</ref></li>
<li>The statement of [[Riesz's lemma]] holds when the real number<ref group=note>The statement of [[Riesz's lemma]] involves only one real number, which is denoted by <math>\alpha</math> in the article on Riesz's lemma. The lemma always holds for all real <math>\alpha < 1.</math> But for a Banach space, the lemma holds for all <math>\alpha \leq 1</math> if and only if the space is reflexive.</ref> is exactly <math>1.</math>{{sfn|Diestel|1984|p=6}} Explicitly, for every closed proper vector subspace <math>Y</math> of <math>X,</math> there exists some vector <math>u \in X</math> of unit norm <math>\|u\| = 1</math> such that <math>\|u - y\| \geq 1</math> for all <math>y \in Y.</math> 
* Using <math>d(u, Y) := \inf_{y \in Y} \|u - y\|</math> to denote the distance between the vector <math>u</math> and the set <math>Y,</math> this can be restated in simpler language as: <math>X</math> is reflexive if and only if for every closed proper vector subspace <math>Y,</math> there is some vector <math>u</math> on the [[unit sphere]] of <math>X</math> that is always at least a distance of <math>1 = d(u, Y)</math> away from the subspace. 
* For example, if the reflexive Banach space <math>X = \Reals^3</math> is endowed with the usual [[Euclidean norm]] and <math>Y = \Reals \times \Reals \times \{0\}</math> is the <math>x-y</math> plane then the points <math>u = (0, 0, \pm 1)</math> satisfy the conclusion <math>d(u, Y) = 1.</math> If <math>Y</math> is instead the <math>z</math>-axis then every point belonging to the unit circle in the <math>x-y</math> plane satisfies the conclusion. 
</li>
<li>Every continuous linear functional on <math>X</math> attains its supremum on the closed unit ball in <math>X.</math><ref>Theorem 1.13.11 in {{harvtxt|Megginson|1998|p=125}}.</ref> ([[James' theorem]])</li>
</ol>

Since norm-closed [[Convex set|convex subsets]] in a Banach space are weakly closed,<ref>Theorem 2.5.16 in {{harvtxt|Megginson|1998|p=216}}.</ref> 
it follows from the third property that closed bounded convex subsets of a reflexive space <math>X</math> are weakly compact.  Thus, for every decreasing sequence of non-empty closed bounded convex subsets of <math>X,</math> the intersection is non-empty.  As a consequence, every continuous [[convex function]] <math>f</math> on a closed convex subset <math>C</math> of <math>X,</math> such that the set
<math display="block">C_t = \{ x \in C \,:\, f(x) \leq t \}</math>
is non-empty and bounded for some real number <math>t,</math> attains its minimum value on <math>C.</math>

The promised geometric property of reflexive Banach spaces is the following: if <math>C</math> is a closed non-empty [[Convex set|convex]] subset of the reflexive space <math>X,</math> then for every <math>x \in X</math> there exists a <math>c \in C</math> such that <math>\|x - c\|</math> minimizes the distance between <math>x</math> and points of <math>C.</math>  This follows from the preceding result for convex functions, applied to<math>f(y) + \|y - x\|.</math>  Note that while the minimal distance between <math>x</math> and <math>C</math> is uniquely defined by <math>x,</math> the point <math>c</math> is not.  The closest point <math>c</math> is unique when <math>X</math> is uniformly convex.

A reflexive Banach space is [[Separable space|separable]] if and only if its continuous dual is separable.  This follows from the fact that for every normed space <math>Y,</math> separability of the continuous dual <math>Y^{\prime}</math> implies separability of <math>Y.</math><ref>Theorem&nbsp;1.12.11 and Corollary&nbsp;1.12.12 in {{harvtxt|Megginson|1998|pp=112–113}}.</ref>