$V$ is isomorphic to $U$ as normed vector space. $U$ is Banach if and only if $V$ is Banach. I don't know exactly, it seems easy at first look, but I have deep problem with the way I have to write the answer. I need your helps. Thank you.
2026-04-13 16:03:08.1776096188
V is isomorphic to U. U is Banach iff V is Banach
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If they are just isomorphic as vector spaces (i.e., there is a bijective linear map between them) this is wrong. Consider e.g. the Banach space $(\ell_2,\|\cdot\|_2)$ and a strictly coarser (smaller) norm on it, e.g. $\|x\|_\infty =\sup\lbrace |x_n|:n\in\mathbb N\rbrace$.
On the other hand, if they are isomorphic as normed spaces you can easily verify that the isomorphism preserves Cauchy as well as convergent sequences in both directions (you will need that continuous linear maps between normed spaces are uniformly continuous).