Exercise 8, Appendix 0

Let E and F be two real Hilbert spaces. There exists on $E \otimes F$ a Hausdorff pre-Hilbert space structure such that $(x_1 \otimes y_1 | x_2 \otimes y_2) = (x_1 | x_2) \cdot (y_1 | y_2)$ for $x_1, x_2$ in E and $y_1, y_2$ in F. One denotes by $E \otimes_2 F$ the Hilbert space completion of $E \otimes F$.

a) If $E'$ (resp. $F'$) is a closed linear subspace of E (resp. F), show that $E' \otimes_2 F'$ may be identified with the closed linear subspace of $E \otimes_2 F$ generated by the elements $x \otimes y$ such that $x \in E'$ and $y \in F'$.

b) Suppose that E (resp. F) is the hilbertian sum of a family $(E_\alpha)_{\alpha \in A}$ (resp. $(F_\beta)_{\beta \in B}$) of closed linear subspaces (TVS, V, §2, No. 2, Def. 2). Show that $E \otimes_2 F$ is the hilbertian sum of the family $(E_\alpha \otimes_2 F_\beta)_{(\alpha, \beta) \in A \times B}$ of closed linear subspaces.

c) If $(e_i)_{i \in I}$ (resp. $(f_j)_{j \in J}$) is an orthonormal basis of E (resp. F), then the family $(e_i \otimes f_j)_{(i, j) \in I \times J}$ is an orthonormal basis of $E \otimes_2 F$.

d) Let G be a real Hilbert space. Define canonical isomorphisms of $E \otimes_2 F$ onto $F \otimes_2 E$ and of $(E \otimes_2 F) \otimes_2 G$ onto $E \otimes_2 (F \otimes_2 G)$.

e) Let $E_1$ and $F_1$ be two real Hilbert spaces, and $u : E \to E_1, v : F \to F_1$ continuous linear mappings. Show that $u \otimes v$ may be extended by continuity to a continuous linear mapping $u \otimes_2 v$ of $E \otimes_2 F$ into $E_1 \otimes_2 F_1$, and that $\|u \otimes_2 v\| = \|u\| \cdot \|v\|$.