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Well I'm reading about Reflexive spaces those days and I would like to see a proof for two different claims.

The first claim is that a Banach space is reflexive iff every bounded functional attains its maximum on the unit ball. Well, It is well known that a Banach space is reflexive iff its dual is reflexive iff the closed ball is weakly compact. From those equivalent characterizations, one can prove that every bounded sequence has a weakly convergent subsequence, and from this one can show that every bounded functional gets a maximum value on the ball. My question is: how can we prove the other side? Given that every bounded functional attains its maximum value on the closed ball, how can we prove that the space is reflexive in order to get the equivalence of all those characterizations?

Another claim is that if for every point x in the space and for every closed and convex set K, the distance between x and K is attained for some x0 in K, then the space is reflexive. This is a question I found within an old exam paper, therefore I suppose that It is much easier to be proven, but I do not see how. I believe that this must follow from one of the characterizations above.

Thank you in advance.

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    $\begingroup$ The first is a (very) non-trivial result due to R. C. James. A proof is contained in Robert E. Megginson's An Introduction to Banach Space Theory. $\endgroup$ – David Mitra Jan 19 '15 at 20:43
  • $\begingroup$ Thank you for your instant response! For the second question I think that the solution must be much easier! $\endgroup$ – Nick Kolliopoulos Jan 19 '15 at 20:56
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    $\begingroup$ The original paper, I believe, is Robert C. James, Characterizations of reflexivity, Studia Math. 23 (1964), 205-216. $\endgroup$ – David Mitra Jan 19 '15 at 21:04
  • $\begingroup$ You ought to add more line breaks. The question is very hard to read as it stands. Typesetting math in dollar signs also helps. $\endgroup$ – tomasz Jan 19 '15 at 21:18
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As mentioned in the comments, your first claim is a justly famous result of R. C. James.

For the second question, you can use this result to show every continuous linear functional on the space attains its norm on the unit sphere of the space; then appeal to James' theorem. (note "attains its maximum on the unit ball" is the same as saying "attains its norm on the unit ball").

I don't see an easy way to prove the claim...

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