Embedding weakly compact sets into Hilbert space

We give an example of a weakly compact set in a Banach space, which does not embed topologically as a weakly compact subset of Hilbert space. We also show that a weakly compact set embeds in a super-reflexive space iff it embeds in Hilbert space. This work was done while the authors were visiting Ohio State University. We wish to thank the Department of mathematics at Ohio State University for their kind hospitality. The work of the first author was partially supported by NSF Grant MPS-74-24249. The work of the second author is a part of his Ph.D. Thesis prepared at the University of California at Berkeley.