Talk:Eberlein–Šmulian theorem
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Definition of "weakly compact" etc
[edit]I guess that "A is weakly P " means that A has property P with respect to the weak topology on X. Can anyone think of a way of making this clear without spoiling the elegant presentation of this article? Jowa fan (talk) 04:47, 5 July 2011 (UTC)
- Hi Jowa, I think that is an explanation best left to the article on weak topology, which is referenced in the article. 69.60.103.171 (talk) 01:07, 3 December 2011 (UTC)
Application to Sobolev spaces
[edit]The application to Sobolev spaces is not very compelling. Most (reflexive) Sobolev spaces X that arise in PDE are separable, and in this case it is a standard and fairly easy result that bounded sets in X^* are weak-* metrizable (you embed the set homeomorphically into the Hilbert cube). Since it's reflexive, the weak and weak-* topologies coincide and so such sets are also weakly metrizable. So this case of the Eberlein–Šmulian theorem is essentially trivial. The interesting cases, which make the result hard enough to deserve to be a "named" theorem, are when X is not reflexive or not separable. 138.86.122.185 (talk) 18:19, 9 February 2018 (UTC)