Norms for the characterisation of weak*-convergence of bounded sequences
Let \(X\) be a separable normed space with dual space \(X'\) and let \(\sigma=(x_k)\) be a separating sequence in the unit sphere \(S_X,\) that is, the span of \(\{x_k: k\in{\mathbb N}\}\) is dense in \(X.\) The author considers on \(X',\) and this is the main idea of the paper, the norm \(\|\;\|_\sigma\) defined by \[ \|x'\|_\sigma:= \sum_{k=1}^\infty 2^{-k}|x'(x_k)| \] and satisfying \(\|x'\|_\sigma\leq \|x'\|\) (Lemma 1) where \(\|\;\|\) is the usual norm on \(X'.\) Then a bounded sequence \((x_n')\) in \(X'\) converges in the weak*-topology if and only if it is \(\|\;\|_\sigma\)-convergent (Theorem 1). This result enables the author to give elementary proofs of the Krein-Milman theorem (Theorem 2) and some variants of it (in the language of locally convex spaces). Another consequence of it is the result of Schur (Theorem 3) that in \( \ell^1\) the norm-convergence and the weak-convergence of sequences coincide. Theorem 4, a further main result based on the idea to consider \(\|\;\|_\sigma,\) says the in case of a normed space \(X\): (a) \(X\) is separable if and only if there exist a separable normed space \(Y\) and a compact linear operator \(T:Y\to X\) with dense image. (b) If \(T\) is such an operator and \((x_n')\) is a bounded sequence in \(X',\) then \((x_n')\) converges to \(x'\) in \(X'\) if and only if \((T'x_n')\) converges to \(Tx'\) in \(Y\) (where \(T'\) is the adjoint map of \(T\)). Applying Theorem 4, the author generalizes the Riemann-Lebesgue lemma (Lemma 4).
scientific article; zbMATH DE number 1757426