Corollary 2

Tag 0823, permanent. Cite it as [Bourbaki, Tag 0823].

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

Let E be a hilbertian space. Then $\mathscr{L}^1(E) \subset \mathscr{L}^2(E)$ .

The real case reduces to the complex case by the extension of scalars; we can therefore assume that $K = \mathbf{C}$ .

Since $\mathscr{L}^2(E)$ is a vector subspace of $\mathscr{L}(E)$ , it is enough to prove that every continuous and positive endomorphism $v$ of E with finite trace belongs to $\mathscr{L}^2(E)$ . With the notations of cor. 1, we have

\sum_{i \in I} \| v(e_i) \|^2 = \sum_{i \in I} \lambda_i^2 \leq (\sum_i \lambda_i)^2 < + \infty .

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

Cited by

Nothing.

Cites

Nothing.