Exercise 2, § 4

a) Let X be the interval $[0, 1]$ of $\mathbf{R}$; show that if $(a_k)_{1 \leqslant k \leqslant n}$ is any finite sequence of pairwise distinct elements of X, then the n functions $|x - a_k|$ ($1 \leqslant k \leqslant n$) are linearly independent. From this, deduce that the continuous function $|x - y|$ defined on $X \times X$ is not of the form $\sum_{i=1}^n u_i(x)v_i(y)$, where the $u_i$ and $v_i$ are continuous.

b) For every measure $\mu \in \mathscr{M}(X; \mathbf{C})$, set $f_\mu(y) = \int |x - y|\ d\mu(x)$. Show that the linear mapping $\mu \mapsto f_\mu$ of $\mathscr{M}(X; \mathbf{C})$ into $\mathscr{C}(X; \mathbf{C})$ is injective, has for image a dense linear subspace L of $\mathscr{C}(X; \mathbf{C})$, and is continuous but not bicontinuous for the normed space topologies on $\mathscr{M}(X; \mathbf{C})$ and L. (Observe that L contains the constants and the piecewise linear functions.)

c) Let $u_i$ ($1 \leqslant i \leqslant m$) be any m functions in $\mathscr{C}(X; \mathbf{C})$ and let V be the linear subspace of $\mathscr{M}(X; \mathbf{C})$ orthogonal to the $u_i$, which is therefore of codimension $\leqslant m$ in $\mathscr{M}(X; \mathbf{C})$. Show that if B is the set of measures $\mu$ on X such that $\| \mu \| \leqslant 1$, there exist measures $\mu \in B$ and $\nu \in V$ such that $\int f_\mu(y)\ d\nu(y)$ is arbitrarily large (make use of b)). From this, deduce that if B, $\mathscr{M}(X; \mathbf{C})$ and $\mathscr{M}(X \times X; \mathbf{C})$ are equipped with the vague topology, then the mapping $(\mu, \nu) \mapsto \mu \otimes \nu$ of $B \times \mathscr{M}(X; \mathbf{C})$ into $\mathscr{M}(X \times X; \mathbf{C})$ is not continuous.