$l^2(\mathbb Z)$ is complete in a different metric.

Question

I want to show that $l^2(\mathbb Z)$ is complete in the metric induced by the inner product $(x,y)=\sum_{j\in \mathbb Z}\lambda_jx_j\bar y_j$, where $(\lambda_j)$ is a real sequence with $0<\lambda_j<1$ for each j. I'm able to show that for a Cauchy sequence in this metric, it has a limit, but I'm stuck showing the limit is in $l^2(\mathbb Z)$. Any help is appreciated.

Answer 1

This is false. Say $P$ is the set of sequences with only finitely many non-zero entries. The completion of $P$ in your new metric is the space of sequences with $\sum\lambda_j|x_j|^2<\infty$; if $\inf\lambda_j=0$ this is strictly larger than $\ell^2$.