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A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Compact operators on $\ell^1$

Let $T$ be a compact symmetric operator on $\ell^2$ and $T\vert_{\ell^1}$ be bounded on $\ell^1$. Are there any non-trivial conditions that $T\vert_{\ell^1}$ is compact as well (for example would $T$ …
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