Example

— Let E be a hilbertian space and I a set of indices. Let $\ell^2_E(I)$ denote the external hilbertian sum of the family $(E_i)_{i \in I}$ where $E_i = E$ for all $i \in I$. In other words, $\ell^2_E(I)$ is the space of all families $x = (x_i)_{i \in I}$ of elements of E such that $\sum_{i \in I} \|x_i\|^2 < +\infty$, endowed with the scalar product $\langle x|y \rangle = \sum_{i \in I} \langle x_i|y_i \rangle$ (space of square summable families of elements of E indexed by I). We put $\ell^2(I) = \ell^2_K(I)$.