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We seem to be talking at cross purposes so let me try again (I do want to help). It seems to me that the difficulty lies in the fact that you are talking about quasi-completeness, me about compactness of unit balls. But in the context of your question, these coincide. This is because the unit ball of a Banach space (resp. of a dual Banach space) is precompact for the weak (resp. for the weak star uniformity) so that they are compact for these topologies if and only they are complete for the associated uniformities. Please accept this in the spirit it is intended and not as "teaching".
It is indeed László. This article probably hasn't been published---it is more a memo. But you should be able to find it simply by googling author plus the title I gave. Anyway it worked for me.
Have managed to dig up Grotndieck's text "Espaces vectoriels topologiques". The result mentioned is on p. 277. By the way, if $E$ is a Banach space, then it bidual is the linear span of the bipolar of its unit ball in the algebraic dual of its dual. The latter is, by the bipolar theorem, its closure in the weak topology induced by $E'$. From this, everything follows.
I don't have access to the texts now so can't give you page and theorem numbers. The theorem of Alaoglu (sometimes Banach-Alaoglu) states that the unit ball of the dual of a Banach space is weak star compact. In the case of reflexivity, this gives what you require.
The above result is contained in an appendix to a longer paper on pde's by J. B. Diaz published in "Theory of Distributions" (Lisbon, 1964)--- the whole volume is available online. It requires no structure on the underlying set---in contrast to the Kakutani FPT, at least in the versions known to me.
