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A colleague of mine showed me the following problem:

Let H be a Hilbert space, show that every compact operator $T:H\rightarrow H$ is the limit of a succesion of compact operators of finite rank(That is, their range is a finite dimension vectorial space)

Any help would be greatly appreciated :)

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  • $\begingroup$ You mean finite rank: their range is a finite dimensional space. $\endgroup$ – Robert Israel May 8 at 15:55
  • $\begingroup$ @Robert Israel, I edited that, thank you $\endgroup$ – miraunpajaro May 8 at 17:50
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Hint: consider the polar decomposition of $T$.

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  • $\begingroup$ I'll try that, I don't know anything about polar decomposition $\endgroup$ – miraunpajaro May 8 at 17:49

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