Example 2

Tag 081W, permanent. Cite it as [Bourbaki, Tag 081W].

Topological Vector Spaces, chapter V, § 4. Some classes of operators in hilbertian spaces

\* Let X and Y be two Hausdorff topological spaces, endowed respectively with positive measures $\mu$ and $\nu$ . We can show that the Hilbert-Schmidt mappings from $\mathscr{L}^2(X)$ into $\mathscr{L}^2(Y)$ correspond bijectively to classes of square integrable functions on $Y \times X$ ; to the class of a function $N \in \mathscr{L}^2(Y \times X, \nu \otimes \mu)$ corresponds the mapping $u_N$ given by

(u_N f)(y) = \int_X N(y, x) f(x) \, d\mu(x)

for

\nu

-almost all

y \in Y

and

f \in \mathscr{L}^2(X, \mu)

. We have

\| u_N \|_2^2 = \int_X \int_Y |N(y, x)|^2 \, d\mu(x) \, d\nu(y) .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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