Polynomials, symmetric multilinear forms and weak compactness (Q1768347)

In the beginning, the author defines a continuous 2-homogeneous polynomial on the real Hilbert space \(\ell_2\) in order to show that not every continuous polynomial on a reflexive Banach space attains its norm. \newline The main results are the following versions of James' weak compactness theorem. Theorem. Let \(A\) be a bounded, closed and convex subset of a (real) Banach space \(E\). Then \(A\) is weakly compact iff there are an \(n \in {\mathbb N}\) and \(x_1^\ast, \ldots, x_n^\ast \in E^\ast\) such that for every \(x^\ast \in E^\ast\) the modulus of the polynomial \( x\mapsto x_1^\ast(x) \cdots x_n^\ast(x)x^\ast(x)\), \(x \in E\), attains its supremum if it is restricted to \(A\) and if \(A \not\subseteq \bigcup_{j=1}^n\) ker \(x_j^\ast\). It follows immediately from this that a Banach space \(E\) is reflexive iff there are an \(n \in {\mathbb N}\) and \(x_1^\ast, \ldots, x_n^\ast \in E^\ast \setminus \{0\}\) such that for every \(x^\ast \in E^\ast\) the polynomial \( x\mapsto x_1^\ast(x) \cdots x_n^\ast(x)x^\ast(x)\), \(x \in E\), attains its norm. In the last section, the author shows that the above polynomial can be replaced by the associated symmetric \((n+1)\)-linear mapping, but the proof is more complicated.