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The famous James theorem states that:

Theorem. Let $X$ be a (Hausdorff separated) locally convex space (LCS for short) with topological dual $X^*$ and let $B\subset X$ be weakly-closed. If $X$ is complete and every $x^*\in X^*$ attains a maximum on $B$ then $B$ is weakly-compact.

(see Theorem 6, page 139 in MR0165344 (29 #2628) James, Robert C. Weakly compact sets. Trans. Amer. Math. Soc. 113 1964 129–140. 46.10)

My question is about the validity of the following converse (in a contra-positive form)

Conjecture. Let $X$ be a LCS. If $X$ is not complete then there exists a weakly-closed set $B\subset X$ such that every $x^*\in X^*$ attains a maximum on $B$ and $B$ is not weakly-compact.

James also built an example of a normed linear space that is not complete but each continuous linear functional attains its supremum on the unit ball. This is just one example that verifies the conjecture. The question is whether the conjecture holds for every non-complete LCS.

I would appreciate any additional information you have on the subject.

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I doubt this could be true. For example, let $X$ be $c_{00}$ (the set of eventually vanishing sequences) with the $\ell_1$ norm. The dual is $\ell^\infty$. Could there be a set $B\subset c_{00}$ which is not contained in a finite-dimensional subspace, and on which every element $x\in \ell_\infty$ attains its maximum? I can't imagine. – user53153 Dec 24 '12 at 6:40

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