Exercise 1, § 1

Let A be an infinite set. a) Let E be the Banach space $\overline{\mathscr{K}(A)}$ over $\mathbf{R}$, consisting of all families $x = (x_\alpha)_{\alpha \in A}$ of real numbers such that $\alpha \mapsto x_\alpha$ tends to 0 with respect to the filter of complements of finite subsets of A, and endowed with the norm $\|x\| = \sup_{\alpha \in A} |x_\alpha|$ (when $A = \mathbf{N}$, this space is denoted by $c_0$ or $c_0(\mathbf{N})$). Show that every continuous linear form on E can be written in a unique way as $x \mapsto \sum_{\alpha \in A} u_\alpha x_\alpha$, where $(u_\alpha)_{\alpha \in A}$ is a family such that $\sum_{\alpha \in A} |u_\alpha| < +\infty$; then the dual $E'$ of E can be identified (as a non topological vector space) with the space $\ell^1(A)$ (I, p. 4, Example). b) Let F be the Banach space $\ell^1(A)$ (I, p. 4, Example) (when $A = \mathbf{N}$, also denoted by $\ell^1$). Show that every continuous linear form on F can be written in a unique way as $x \mapsto \sum_{\alpha \in A} u_\alpha x_\alpha$, where $(u_\alpha)_{\alpha \in A}$ is a bounded family of real numbers; then the dual $F'$ of F can be identified (as a non topological vector space) with the space $\mathscr{B}(A) = \ell^\infty(A)$. c) Let B be an arbitrary set, $(c_{\alpha \beta})_{(\alpha, \beta) \in A \times B}$ an arbitrary family of numbers $\geqslant 0$. Let G be the vector space of all families $x = (x_\alpha)_{\alpha \in A}$ of real numbers such that, for every $\beta \in B$, we have $p_\beta(x) = \sum_{\alpha \in A} c_{\alpha \beta} |x_\alpha| < +\infty$; the $p_\beta$ are semi-norms on G. In order that G, with the topology defined by this family of semi-norms, be Hausdorff, it is necessary and sufficient that, for every $\alpha \in A$, there exists at least one $\beta \in B$ such that $c_{\alpha \beta} > 0$. Show that then G is complete, and that every continuous linear form on G can be written uniquely as $x \mapsto \sum_{\alpha \in A} u_\alpha x_\alpha$, where $(u_\alpha)_{\alpha \in A}$ is a family of real numbers satisfying the following condition; there exists a finite number of indices $\beta_i \in B$ ($1 \leqslant i \leqslant n$) and a number $a > 0$ such that $|u_\alpha| \leqslant a \cdot c_{\alpha \beta_i}$ for all $\alpha \in A$ and $1 \leqslant i \leqslant n$. Prove the converse, and extend these results to the case where the field of scalars is $\mathbf{C}$.