James's theorem

Remarks

The direction "reflexive implies norm attaining" is short. In a reflexive space the closed unit ball is weakly compact, and $x \mapsto \abs{f(x)}$ is weakly continuous, so it attains its maximum $\norm{f}$. The converse is the deep part, and its proof by James is long and combinatorial.

The theorem gives a concrete test for non-reflexivity. On $c_0$, the functional $f(x) = \sum_{n \ge 1} 2^{-n} x_n$ has norm $\sum_n 2^{-n} = 1$. For $x \in c_0$ with $\norm{x}_\infty \le 1$, some coordinate has $\abs{x_n} < 1$, so $\abs{f(x)} < 1$. So $f$ does not attain its norm, and $c_0$ is not reflexive.

Going deeper

For a student

James's theorem is used in optimization and convex analysis, where the existence of maximizers of linear functionals on the unit ball matters. It contrasts with the Bishop–Phelps theorem, which says that in every Banach space the norm-attaining functionals are dense in $X^*$. It sits beside the Eberlein–Šmulian theorem and the Goldstine theorem as a main characterization of weak compactness and reflexivity.