I know that $\ell_2$ with respect to $\Vert \cdot \Vert_2$ norm is complete? I can't figure out for this set.
2026-04-01 14:56:16.1775055376
Is $(\ell_1 , \Vert \cdot \Vert_2)$ a complete space?
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Consider the sequence $(f_n) \in \ell_1^{\mathbb N}$ defined by $$\begin{cases} f_n(k)= 1/k & \mbox{for} & k\le n\\ 0 &\mbox{for} & k>n \end{cases}$$
$(f_n)$ is a Cauchy sequence regarding $\Vert \cdot \Vert_2$ as the sequence $\sum 1/k^2$ converges. However $(f_n)$ doesn’t converges towards an element of $\ell^1$ as $\sum 1/k$ diverges.
Conclusion: $(\ell_1 , \Vert \cdot \Vert_2)$ is not complete