Example 5

Tag 07XL, permanent. Cite it as [Bourbaki, Tag 07XL].

Topological Vector Spaces, chapter V, § 1. Prehilbertian spaces and hilbertian spaces

\* Let D be the open disc with centre 0 and radius 1 in $\mathbf{C}$ . The Hardy space $H^2(D)$ consists of all holomorphic functions $f : D \to \mathbf{C}$ for which

\sup_{0 < R < 1} \int_0^1 |f(R \cdot e(\theta))|^2 d\theta < + \infty .

f_1

and

f_2

belong to

H^2(D)

, the limit

\langle f_1 | f_2 \rangle = \lim_{R \to 1} \int_0^1 \overline{f_1(R \cdot e(\theta))} \cdot f_2(R \cdot e(\theta)) \, d\theta

exists; the mapping

(f_1, f_2) \mapsto \langle f_1 | f_2 \rangle

assigns a complex hilbertian space structure to the vector space

H^2(D)

For a function $f : D \to \mathbf{C}$ to belong to $H^2(D)$ , it is necessary and sufficient that there exists a sequence $(a_n)_{n \in \mathbf{N}}$ of complex numbers such that $\sum_{n=0}^\infty |a_n|^2 < + \infty$ and that

f(z) = \sum_{n=0}^\infty a_n z^n

for all

z \in D

. Then we have

\|f\|^2 = \sum_{n=0}^\infty |a_n|^2

which gives an isomorphism from

H^2(D)

onto the hilbertian space

\ell^2

(V, p. 4).

Every Hausdorff prehilbertian space is isomorphic to an everywhere dense subspace of a hilbertian space determined up to an isomorphism. Precisely :

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

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