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Let $X,Y$ be Banach Spaces, and let $T\in K(X,Y)$ be a compact operator from $X$ to $Y$. I have to prove that $T(X)$ is closed in Y if, and only if, $\dim(T(X))<\infty$.

Can anybody help me with this proof, please? There is surely some property I haven't thought about, but I'm getting really weird right now... Thank you!

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Let $Z=T(X)$. Then $Z$ is also a Banach space, as a closed subspace of a Banach space, and $T:X\to Z$ is onto, and hence open, due to Open Mapping Theorem. If $Z$ were infinite dimensional, then $T$ would not be compact, as open sets in infinite dimensional spaces are not pre-compact.

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    $\begingroup$ How can I see that open sets cannot be precompact in infinite dimensional spaces? $\endgroup$ – Marc Jan 16 '14 at 22:32
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Hint: Try the open mapping theorem.

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    $\begingroup$ Ok, I just proved that T must be open as it is surjective if you consider it as an operator from $X$ to $T(X)$, and $T(X)$ is a Banach Space as it is closed in Y. But how can I go on with it? Because $T(B_X)$ should then be a open neighbourhood of $0$, but if $dim(T(X))<\infty$ this cannot be, I think. So I would have proved that the dimension is not finite. $\endgroup$ – Marc Jan 15 '14 at 15:41
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    $\begingroup$ It will be a neighbourhood, but not an open one. This is not a problem. But it is also compact by assumption, so it follows that the unit ball of $T(X)$ is compact. Does that ring a bell? $\endgroup$ – Harald Hanche-Olsen Jan 15 '14 at 16:16
  • $\begingroup$ It only can be compact if the dimension of $T(X)$ is finite. But how do we know that the unit ball in $T(X)$ is compact? I'm not seeing something that's obvious, sure... $\endgroup$ – Marc Jan 15 '14 at 16:57
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Hint: an infinite dimensional Banach space is never $\sigma$-compact (by Baire category theorem).

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