Short proof of sequential Banach Alaoglu for Hilbert spaces

Question

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.

score 1 · Accepted Answer · answered Jan 28 '15 at 10:50

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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.

answered Jan 28 '15 at 10:50

Bananach

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

Short proof of sequential Banach Alaoglu for Hilbert spaces

1 Answers1