Exercise 1, § 1

a) Let E be a Banach space over K. There exists a norm $x\mapsto \|x\|$ on E, defining the topology of E, such that $\|x\| \in G$ for all $x\in E$. Such a norm will be called round. If the valuation of K is not discrete, then every norm defining the topology of E is round.

b) Let E and F be Banach spaces over K. The space $\mathscr{L}(E; F)$ is endowed with the norm $\|u\|=$ sup$_{x\in E-\{0\}}\|u(x)\|/\|x\|$. The space $\mathscr{L}(E; F)$ is a Banach space over K. If the norm of E is round, then $\|u\|=$ sup$_{\|x\|\leqslant 1}\|u(x)\|$. If the norm of F is also round, then the norm of $\mathscr{L}(E; F)$ is round. We shall write $E'=\mathscr{L}(E; K)$, and shall say that $E'$ is the dual space of E.

c) Let I be a set, F a Banach space over K and $\mathscr{C}_0(I,F)$ the K-vector space of functions $f: I\rightarrow F$ such that $\|f(i)\|$ tends to 0 according to the filter of the complements of the finite subsets of I. Endowed with the norm $\|f\|=$ sup$_{i\in I}\|f(i)\|$, the space $\mathscr{C}_0(I,F)$ is a Banach space over K, whose norm is round if that of F is. We write $\mathscr{C}_0(I) =\mathscr{C}_0(I,K)$.

d) Let I be a set. For every $i\in I$, we write $\varphi_i\in \mathscr{C}_0(I)$ for the characteristic function of $\{i\}$. Let F be a Banach space over K. The mapping $u\mapsto (u(\varphi_i))_{i\in I}$ is an isomorphism of the Banach space $\mathscr{L}(\mathscr{C}_0(I); F)$ into the Banach space $\mathscr{B}(I; K)$ (EVT, I, p. 4, example ; cf. EVT, I, p. 23, exercise 7).

e) Suppose that the valuation of K is discrete. Let E be a Banach space over K whose norm is round. There exists a set I such that E is isometrically isomorphic to the Banach space $\mathscr{C}_0(I) ($cf. EVT, I, p. 23, exercise 7).

f) Suppose that the valuation of K is discrete. Let E be a Banach space over K endowed with a round norm and F a closed vector subspace of E. There exists a continuous projector of E whose image is F and which is of norm $\leqslant 1$.