- #1

Fredrik

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## Main Question or Discussion Point

This is probably easy. It's really annoying that I don't see how to do this...

A finite rank operator (on a Hilbert space) is a bounded (linear) operator such that its range is a finite-dimensional subspace. I want to show that if T has finite rank, than so does T*.

I'm thinking that the theorem that says [tex]\operatorname{ker} T=(\operatorname{ran} T^*)^\perp[/tex] should be relevant, but I don't see how to use it. Feel free to use [itex]\|T\|=\|T^*\|[/itex] too if that helps.

Uh, maybe I should have posted this in homework. It's not homework, but a textbook-style question. Moderators, feel free to move it if you want to.

A finite rank operator (on a Hilbert space) is a bounded (linear) operator such that its range is a finite-dimensional subspace. I want to show that if T has finite rank, than so does T*.

I'm thinking that the theorem that says [tex]\operatorname{ker} T=(\operatorname{ran} T^*)^\perp[/tex] should be relevant, but I don't see how to use it. Feel free to use [itex]\|T\|=\|T^*\|[/itex] too if that helps.

Uh, maybe I should have posted this in homework. It's not homework, but a textbook-style question. Moderators, feel free to move it if you want to.

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