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I have to prove that:

If $X,Y$ be Banach Spaces, and $T\in B(X,Y)$ is a compact operator, then $T(X)$ 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!

Yiorgos S. Smyrlis
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Marc
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3 Answers3

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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.

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

Harald Hanche-Olsen
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    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. – Marc Jan 15 '14 at 15:41
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    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? – Harald Hanche-Olsen Jan 15 '14 at 16:16
  • 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... – 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).

tomasz
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