Example 2

\* Let $X$ be a Hausdorff topological space, and $\mu$ a positive measure on $X$. Let $E$ be a hilbertian space. We can identify $L^2(X, \mu) \hat{\otimes}_2 E$ with $L^2_E(X, \mu)$ in a canonical way : if $f$ is the class of the square integrable scalar function $f$ on $X$, and if $a$ belongs to $E$, then $f \otimes a$ is the class of the function $x \mapsto f(x).a$ with values in $E$.

Let $Y$ be a Hausdorff topological space and $\nu$ a positive measure on $Y$. In an analogous manner, we can identify the hilbertian spaces $L^2(X, \mu) \hat{\otimes}_2 L^2(Y, \nu)$ and $L^2(X \times Y, \mu \otimes \nu)$; then $f \otimes g$ can be identified with the class of the function $(x, y) \mapsto f(x)g(y)$ on $X \times Y$. \*