# Weighted $\ell^2$ space is Hilbert

This is my exam's question that I could not solve it. Please help me to undrestand how to solve it: let $\{w_n\}$ is a positive real numbers sequence, and let $$\ell^2(w):=\left\{\{x_n\}:x_n \in R ,\sum_{n=1}^\infty w_n |x_n|^2< \infty\right\}.$$ 1) I needed to show that $\ell^2(w)$ is a Hilbert space under inner product $$\langle x,y\rangle:= \sum_{n=1}^\infty w_n x_n y_n$$ for all $x=\{x_n\}\in \ell^2(w),y=\{y_n\}\in \ell^2(w)$

2) I needed to find a positive linear functional $$f\colon\ell^2(w)\rightarrow R$$ and find it's norm.

Thanks.

• For the first question, do you know what you have to check? – Davide Giraudo Sep 29 '13 at 17:12
• yes,I need to show that the defined inner product is well define and is real an inner product, then I should show every couchy sequence is converge, but how to do it,I really do not know – Rosa Sep 29 '13 at 17:41

1) Well-definiteness of the inner product is given by classical Cauchy-Schwarz inequality. Bilinearity should not be problematic, and positive definiteness uses the fact that the terms of $w$ are positive.
Now it remains to prove completeness. This can be done similarly as the case $w_n=1$ for each $n$.