Exercise 1, § 2

Let S be an arbitrary infinite set. a) Show that the smallest cardinal of any total set in the normed space $\mathscr{B}(S)$ of bounded mappings of S in $\mathbf{R}$ (I, p. 4, Example) is equal to $2^{\mathrm{Card}(S)}$ (consider the set of characteristic functions of subsets of S and note that there exists an enumerable set everywhere dense in $\mathbf{R}$). b) Show that the smallest cardinal of any total set in the normed space $\ell^1(S)$ (I, p. 4, Example) is equal to Card(S).