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.