Exercise 1, § 4

Let $E$ and $F$ be two Hilbert spaces of infinite dimension and of countable type, $(a_n)$ an orthonormal basis of E, $(b_n)$ an orthonormal basis of F. a) Let $u$ be a continuous linear mapping of E into F; put $u(a_n) = \sum_m \alpha_{mn} b_m$. Now prove that one has $\sum_n |\alpha_{mn}|^2 \leq \|u\|^2$ and $\sum_m |\alpha_{mn}|^2 \leq \|u\|^2$ whatever $m$ and $n$ may be. b) Give an example of a double sequence $(\alpha_{mn})$ such that $\sum_m |\alpha_{mn}|^2 \leq 1$ for every $n$ and $\sum_n |\alpha_{mn}|^2 \leq 1$ for every $m$, but such that there exists no continuous linear mapping $u$ of E into F such that $\langle u(a_n)|b_m \rangle = \alpha_{mn}$ for every pair of integers $(m, n)$. (Prove that if $I \subset \mathbf{N}$ is a set of $p$ integers, and if $V_p$ (resp. $W_p$) is the subspace of E (resp. F) generated by the $a_n$ (resp. $b_n$) such that $n \in I$, there exists a linear mapping $u_p$ of $V_p$ onto $W_p$ such that $\langle u_p(a_n)|b_m \rangle = \frac{1}{\sqrt{p}}$ for $m \in I$ and $n \in I$, and that one has $\|u_p\| \geq \sqrt{p}$.)