Editing Banach–Alaoglu theorem (section)

===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>