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Let $X$ be a separable space with infinite dimension, let $(\cdot,\cdot)$ and $\|\cdot \|$ be the scalar product and the norm of $X$, and $\{e_n\}_n$ be an orthonormal basis of $X$. We define $\|x\|_0^2=\sum_{n=1}^\infty\frac{|(x,e_n)|^2}{n^2}.$

What I already proved is that $\|\cdot \|_0$ is a norm and that the set $\{x\in X : \|x\|\leq 1\}$ is compact in $(X,\|\cdot\|_0)$; I am then asked to prove that $(X,\|\cdot\|_0)$ is not complete. So I focused on trying to find a Cauchy sequence in $(X,\|\cdot\|_0)$ not converging to an element in this space. By the previous point such a sequence $\{x_n\}_n$ should not be bounded with respect to the norm $\|\cdot\|$, but up 'till now I was not able to find such an example.

Can anybody help me?

Regards

-Guido-

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I'll assume that $X$ is a Hilbert space. Take $x_n=e_1+e_2+\cdots +e_n$. Using the fact that $\sum\limits_{n=1}^\infty {1\over n^2}$ converges, you can show that $(x_n)$ is Cauchy in $(X,\Vert\cdot\Vert_0)$. But $(x_n)$ cannot converge, since the only "candidate" for what it would converge to is $e_1+e_2+\cdots$.

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    Another way to show that $x_n$ does not converge might be showing that it is unbounded w.r.t original norm $\lVert\cdot\rVert$.2012-05-05