Editing Banach–Alaoglu theorem (section)

==Consequences==

===Consequences for normed spaces===

Assume that <math>X</math> is a [[normed space]] and endow its continuous dual space <math>X^{\prime}</math> with the usual [[dual norm]]. 

<ul>
<li>The closed unit ball in <math>X^{\prime}</math> is weak-* compact.{{sfn|Narici|Beckenstein|2011|pp=225-273}}  So if <math>X^{\prime}</math> is infinite dimensional then its closed unit ball is necessarily {{em|not}} compact in the norm topology by [[F. Riesz's theorem]] (despite it being weak-* compact).
</li>
<li>A [[Banach space]] is [[Reflexive space|reflexive]] if and only if its closed unit ball is <math>\sigma\left(X, X^{\prime}\right)</math>-compact; this is known as [[James' theorem]].{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
<li>If <math>X</math> is a [[reflexive Banach space]], then every bounded sequence in <math>X</math> has a weakly convergent subsequence. 
(This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of <math>X</math>; or, more succinctly, by applying the [[Eberlein–Šmulian theorem]].) 
For example, suppose that <math>X</math> is the space [[Lp space]] <math>L^p(\mu)</math> where <math>1 < p < \infty</math> and let <math>q</math> satisfy <math>\frac{1}{p} + \frac{1}{q} = 1.</math>
Let <math>f_1, f_2, \ldots</math> be a bounded sequence of functions in <math>X.</math> 
Then there exists a subsequence <math>\left(f_{n_k}\right)_{k=1}^{\infty}</math> and an <math>f \in X</math> such that
<math display=block>\int f_{n_k} g\,d\mu \to \int f g\,d\mu \qquad \text{ for all } g \in L^q(\mu) = X^{\prime}.</math>
The corresponding result for <math>p = 1</math> is not true, as <math>L^1(\mu)</math> is not reflexive.</li>
</ul>

===Consequences for Hilbert spaces===

<ul>
<li>In a Hilbert space, every bounded and closed set is weakly relatively compact, hence every bounded net has a weakly convergent subnet (Hilbert spaces are [[reflexive space|reflexive]]).</li>
<li>As norm-closed, convex sets are weakly closed ([[Hahn–Banach theorem]]), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact.</li>
<li>Closed and bounded sets in <math>B(H)</math> are precompact with respect to the [[weak operator topology]] (the weak operator topology is weaker than the [[ultraweak topology]] which is in turn the weak-* topology with respect to the predual of <math>B(H),</math> the [[trace class]] operators). Hence bounded sequences of operators have a weak accumulation point. 
As a consequence, <math>B(H)</math> has the [[Heine–Borel theorem|Heine–Borel property]], if equipped with either the weak operator or the ultraweak topology.</li>
</ul>