Definition 1

Let $(E_i)_{i \in I}$ be a family of hilbertian spaces. The hilbertian space E defined in prop. 1 is called the external hilbertian sum of the family $(E_i)_{i \in I}$ and written as $\bigoplus_{i \in I} E_i$ or $\bigoplus_{i \in I} E_i$^1.

Let $f_i$ be the mapping from $E_i$ into E which transforms $z \in E_i$ into an element $(x_k) \in E$ such that $x_k = 0$ for all $k \neq i$ and $x_i = z$; it is clear that $f_i$ is an isomorphism from the hilbertian space $E_i$ onto a closed vector subspace of E. We say that $f_i$ is the canonical mapping from $E_i$ into E and we shall generally identify $E_i$ with its image in E by this isomorphism. With this convention, $E_i$ and $E_k$ are orthogonal in E for $i \neq k$, and E is the closed vector subspace generated by the union of the subspaces $E_i$.

When I is finite, E is the direct sum of the $E_i$; since the canonical projector from E onto $E_i$ is continuous for all $i \in I$, E is also the topological direct sum of the $E_i$ (GT, III, § 6, No. 2, prop. 2). If $I = \{1, n\}$, we also write $E_1 \oplus E_2 \oplus ... \oplus E_n$ instead of $\bigoplus_{i=1}^n E_i$.