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**ejgmath** Let $\displaystyle H$ be the space of sequences $\displaystyle (x(1),x(2),x(3),...)$ such that $\displaystyle \sum^{\infty}_{j=1}\frac{|x(j)|^2}{j^2}<\infty$, with inner product $\displaystyle <x,y>=\sum^{\infty}_{j=1}\frac{x(j)\overline{y(j)} }{j^2}$ and norm $\displaystyle \|x\|=<x,x>^{1/2}$.

I need to show that $\displaystyle H$ is complete with respect to the inner product i.e. a Hilbert Space. I am assuming that $\displaystyle \mathbb{C}$ is complete.

I know that I need to show that every Cauchy sequence in H is convergent. I can see that this follows relatively easily from the definition of H.

So I have taken a sequence $\displaystyle x_{n}(j)$ but for this to be Cauchy $\displaystyle |x_{n}(j)-x_{m}(j)|<\epsilon$. But I can't see how to show that, given that the inner product on $\displaystyle H$ has a $\displaystyle j^2$ below it. Any help would be great. Thanks