Hilbert space

Functional Analysis
Post Reply
Papapetros Vaggelis
Community Team
Posts: 426
Joined: Mon Nov 09, 2015 1:52 pm

Hilbert space

#1

Post by Papapetros Vaggelis »

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

\(\displaystyle{H}\).
Post Reply

Create an account or sign in to join the discussion

You need to be a member in order to post a reply

Create an account

Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute

Register

Sign in

Who is online

Users browsing this forum: No registered users and 1 guest