Consider the following famous theorem by Robert C. James (1964):
Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and only if every continuous linear real-valued functional on $X$ attains its supremum on $C$.
The “only if” part is trivial, while known proofs of the “if” part are notoriously involved. Some texts prove it only for the case in which $X$ is assumed to be separable (see, for example, Holmes, 1975, pp. 157–161). Some sources do not make this extra assumption, yet they tend to be still too involved for my taste and patience (see, for example, Pryce, 1966).
Using Holmes (1975), who treats only the separable case, I can prove the following statement:
If $C\subseteq X$ is non-empty, bounded, weakly closed, but not weakly compact, then there exists a separable norm-closed subspace $Y\subseteq X$ and a continuous real-valued functional $f:Y\to\mathbb R$ on it such that $C\cap Y$ is not empty and $f$ doesn't attain its supremum on $C\cap Y$.
My question is: Do you think the second statement can be used to prove the existence of a continuous real-valued functional $F:X\to\mathbb R$ that doesn't attain its supremum on $C$? Do you think trying extending $f$ onto the whole space by using the Hahn–Banach theorem, or Zorn's lemma, could lead to anything useful?
- James, R. C. (1964): “Weakly Compact Sets,” Transactions of the American Mathematical Society 113, 129–140.
- Holmes, R. B. (1975): Geometric Functional Analysis and Its Applications, New York: Springer-Verlag.
- Pryce, J. D. (1966): “Weak Compactness in Locally Convex Spaces,” Proceedings of the American Mathematical Society 17, 148–155.