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Hilbert space

Functional Analysis
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Papapetros Vaggelis
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Hilbert space

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Post by Papapetros Vaggelis » Tue Nov 07, 2017 11:12 am

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

\(\displaystyle{H}\).
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