Exercise 5, § 1

Let $K$ be a non-discrete valued field, and $S$ be an arbitrary infinite set.

a) Let D = (a_n) be an enumerable infinity of elements of S. For every $\lambda \in K$ such that $|\lambda| \leq 1$, let $u_\lambda$ be the element of the normed space $\mathscr{B}_K(S)$, (I, p. 4, Example) of bounded mappings of S in $\bar{K}$, such that $u_\lambda(a_n) = \lambda^n$ for all $n \in \mathbf{N}$ and $u_\lambda(b) = 0$ for $b \notin D$. Show that the family $(u_\lambda)$ is algebraically independent. b) Deduce that every basis of the vector space $\mathscr{B}_K(S)$ has the same cardinality as $K^S$ (using a) show that the cardinal of every basis of $\mathscr{B}_K(S)$ is at least equal to Card (K); note on the other hand that Card($\mathscr{B}_K(S)$) = Card(K^S) and use A, II, § 2, exerc. 22). c) Show in the same way that every basis of the vector space $\ell_K^1(S)$ has the cardinality of $(K \times S)^{\mathbf{N}}$.