Editing Banach space (section)

===Reflexivity===
{{main|Reflexive space}}

The normed space <math>X</math> is called ''[[Reflexive space|reflexive]]'' when the natural map
<math display=block>\begin{cases} F_X : X \to X'' \\ F_X(x) (f) = f(x) & \text{ for all } x \in X, \text{ and for all } f \in X'\end{cases}</math>
is surjective. Reflexive normed spaces are Banach spaces.

{{math theorem| math_statement = If <math>X</math> is a reflexive Banach space, every closed subspace of <math>X</math> and every quotient space of <math>X</math> are reflexive.}}

This is a consequence of the Hahn–Banach theorem. 
Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space <math>X</math> onto the Banach space <math>Y,</math> then <math>Y</math> is reflexive.

{{math theorem| math_statement = If <math>X</math> is a Banach space, then <math>X</math> is reflexive if and only if <math>X'</math> is reflexive.}}

{{math theorem|name=Corollary | math_statement = Let <math>X</math> be a reflexive Banach space. Then <math>X</math> is [[Separable space|separable]] if and only if <math>X'</math> is separable.}}

Indeed, if the dual <math>Y'</math> of a Banach space <math>Y</math> is separable, then <math>Y</math> is separable. 
If <math>X</math> is reflexive and separable, then the dual of <math>X'</math> is separable, so <math>X'</math> is separable.

{{math theorem| math_statement = Suppose that <math>X_1, \ldots, X_n</math> are normed spaces and that <math>X = X_1 \oplus \cdots \oplus X_n.</math> Then <math>X</math> is reflexive if and only if each <math>X_j</math> is reflexive.}}

Hilbert spaces are reflexive. The <math>L^p</math> spaces are reflexive when <math>1 < p < \infty.</math> More generally, [[uniformly convex space]]s are reflexive, by the [[Milman–Pettis theorem]]. 
The spaces <math>c_0, \ell^1, L^1([0, 1]), C([0, 1])</math> are not reflexive. 
In these examples of non-reflexive spaces <math>X,</math> the bidual <math>X''</math> is "much larger" than <math>X.</math> 
Namely, under the natural isometric embedding of <math>X</math> into <math>X''</math> given by the Hahn–Banach theorem, the quotient <math>X'' / X</math> is infinite-dimensional, and even nonseparable. 
However, Robert C. James has constructed an example<ref>{{cite journal|author = R. C. James|title=A non-reflexive Banach space isometric with its second conjugate space|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=37|pages=174–177|year=1951|issue=3 | doi=10.1073/pnas.37.3.174 | pmc=1063327|pmid=16588998|bibcode=1951PNAS...37..174J |doi-access=free}}</ref> of a non-reflexive space, usually called "''the James space''" and denoted by <math>J,</math><ref>see {{harvtxt|Lindenstrauss|Tzafriri|1977}}, p.&nbsp;25.</ref> such that the quotient <math>J'' / J</math> is one-dimensional. 
Furthermore, this space <math>J</math> is isometrically isomorphic to its bidual.

{{math theorem| math_statement = A Banach space <math>X</math> is reflexive if and only if its unit ball is [[Compact space|compact]] in the [[weak topology]].}}

When <math>X</math> is reflexive, it follows that all closed and bounded [[Convex set|convex subsets]] of <math>X</math> are weakly compact. 
In a Hilbert space <math>H,</math> the weak compactness of the unit ball is very often used in the following way: every bounded sequence in <math>H</math> has weakly convergent subsequences.

Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain [[Infinite-dimensional optimization|optimization problems]]. 
For example, every [[Convex function|convex]] continuous function on the unit ball <math>B</math> of a reflexive space attains its minimum at some point in <math>B.</math>

As a special case of the preceding result, when <math>X</math> is a reflexive space over <math>\R,</math> every continuous linear functional <math>f</math> in <math>X'</math> attains its maximum <math>\|f\|</math> on the unit ball of <math>X.</math> 
The following [[James' theorem|theorem of Robert C. James]] provides a converse statement.

{{math theorem| name = James' Theorem | math_statement = For a Banach space the following two properties are equivalent:
* <math>X</math> is reflexive.
* for all <math>f</math> in <math>X'</math> there exists <math>x \in X</math> with <math>\|x\| \leq 1,</math> so that <math>f(x) = \|f\|.</math>}}

The theorem can be extended to give a characterization of weakly compact convex sets.

On every non-reflexive Banach space <math>X,</math> there exist continuous linear functionals that are not ''norm-attaining''. 
However, the [[Errett Bishop|Bishop]]–[[Robert Phelps|Phelps]] theorem<ref>{{cite journal|last1=bishop|first1=See E.|last2=Phelps|first2=R.|year=1961|title=A proof that every Banach space is subreflexive|journal=Bull. Amer. Math. Soc.|volume=67|pages=97–98|doi=10.1090/s0002-9904-1961-10514-4|doi-access=free }}</ref> states that norm-attaining functionals are norm dense in the dual <math>X'</math> of <math>X.</math>