# Show property of compact operators over Hilbert spaces

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 :)

• You mean finite rank: their range is a finite dimensional space. – Robert Israel May 8 at 15:55
• @Robert Israel, I edited that, thank you – miraunpajaro May 8 at 17:50

Hint: consider the polar decomposition of $$T$$.