Exercise 15, § 5

Let E be a Banach space with a Banach basis $(e_n)$ (exerc. 14); then there exists a unique sequence $(e'_n)$ in the dual $E'$ of E such that the expression of every $x \in E$ in terms of the Banach basis $(e_n)$ can be written as $x = \sum_{n=0}^\infty \langle x, e'_n \rangle e_n$.

a) Let $F_n$ be the closed vector subspace of E generated by the $e_m$ for $m \geq n$. For every $x' \in E'$, let $\|x'\|_n$ denote the norm of the restriction of the linear form $x'$ to $F_n$. Show that, for $(e'_n)$ to be a Banach basis of $E'$, it is necessary and sufficient that for every $x' \in E'$, the sequence $(\|x'\|_n)$ tends to 0. (Consider the transpose $tP_n$ and evaluate the norm $\|tP_n \cdot x' - x'\|$.) In this case, the Banach basis $(e_n)$ is said to be contracting.

b) Suppose that the Banach basis $(e_n)$ is contracting. Show that for every point $x''$ of the bidual $E''$ of E, the sequence of sums $\sum_{m=0}^n \langle e'_m, x'' \rangle e_m$ is bounded in E (consider the transpose $t(tP_n)$ in $E''$). Conversely, for every sequence $(a_n)$ of scalars such that the sequence of sums $\sum_{m=0}^n a_m e_m$ is bounded in E, there exists a unique $x'' \in E''$ such that $\langle e'_n, x'' \rangle = a_n$ for all $n$ (use the compactness of a closed ball in $E''$ for the topology $\sigma(E'', E')$).

c) A Banach basis $(e_n)$ in E is said to be complete if, for every sequence $(a_n)$ of scalars such that the sequence of sums $\sum_{m=0}^n a_m e_m$ is bounded, the series $\sum_{n=0}^\infty a_n e_n$ converges. If $(e_n)$ is a contracting basis of E, the basis $(e'_n)$ of $E'$ is complete (use the compactness of a closed ball in $E'$ for $\sigma(E', E)$).

d) In general, the sequence $(e'_n)$ is a Banach basis of the closed subspace $F'$ of the strong dual $E'$ of E, generated by the $e'_n$, and there is an injective continuous linear mapping $J$ from E into the strong dual $F''$ of $F'$ such that $\langle J.x, z' \rangle = \langle x, z' \rangle$ for all $x \in E$ and all $z' \in F'$. Show that there exists a constant $K > 0$ such that $\|J.x\| \geq K.\|x\|$, and that if the basis $(e_n)$ is complete, then $J$ is an isomorphism from E onto the topological vector space $F''$.

e) Show that for E to be reflexive, it is necessary and sufficient that the basis $(e_n)$ is contracting and complete (use b)).