Proving a space is not complete by finding an absolutely convergent series

Question

For $ v ∈ l^2(\mathbb{N}, F) $ define

$$ ||v||_w=\sum_{k=1}^\infty |v_{[k]}|/2^k$$ as a norm,

Is $ l^2(\mathbb{N}, F) $ with the norm $||v||_w $ a complete space?

I am trying to find a series of vectors which is absolutely convergent, but is not convergent

Answer 1

Hint

i) We have $$\left|\sum a_kb_k \right| \leq \left(\sum |a_k|^2\right)^{1/2}\left(\sum |b_k|^2\right)^{1/2}$$

Use this to show the question is not rubbish.

ii) Now find a sequence $v_n = (a_k)_{k=1}^{k=n}$ (depending on the weight function) such that $v_n$ converges to something in the new norm, but where the limit in not part of $\ell^2$.

To construct the finite sequence, start by looking at $v_n =(a_k)_{k=1}^{k=n}$ where $a_k = 2^k$ for $k=1,\ldots,n$ - this one will not make the job (why?). Now, modify the sequence slightly.

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You can do this!