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## Hilbert space

Functional Analysis
Papapetros Vaggelis
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### Hilbert space

Let $\displaystyle{\left(H,\langle{,\rangle}\right)}$ be a Hilbert space. We set

$\displaystyle{\ell^2(H):=\left\{x:\mathbb{N}\to H\,,\sum_{n=1}^{\infty}||x_{n}||^2<\infty\right\}}$.

and

$\displaystyle{\langle{x,y\rangle}:=\sum_{n=1}^{\infty}\langle{x_n,y_n\rangle}\,,\forall\,x\,,y\in \ell^2(H)}$.

Prove that $\displaystyle{\left(\ell^2(H),\langle{,\rangle}\right)}$ is a Hilbert space which contains

$\displaystyle{H}$.