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Do you know of a short proof of the fact that bounded sequences in Hilbert spaces admit weakly converging subsequences?

If the space is separable, then the common sequential-version proof is what I consider short enough. Also, the common proof of the Heine-Borel-compactness Banach-Alaoglu is short enough. Only, first proving the Heine-Borel-compactness Banach-Alaoglu and then proving Eberlein-Smulian is too long for me. In particular, my problem would be solved if Eberlein-Smulian was evident for Hilbert spaces.

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  • $\begingroup$ oh no, of course. $\endgroup$ – Bananach Jan 28 '15 at 10:49
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As the link in the first comment reveals, one may restrict considerations to the Hilbert-Space generated by the sequence, which is obviously separable.

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    $\begingroup$ Here is the (deleted above) link. $\endgroup$ – David Mitra Jan 28 '15 at 10:51

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