I am studying Functional Analysis and Spectral Theory and I came across this definition in the Wikipedia article Compact Operator that I didn't study in class:
A compact operator on a Hilbert $H$ space is written in the form: $T=\sum_{n\geq 1}{\lambda_{n}\langle f_n,\cdot \rangle g_n}$ where $\{f_{1},f_{2},\ldots \}$ and $\{g_{1},g_{2},\ldots \}$ are orthonormal sets (not necessarily complete), and $\lambda _{1},\lambda _{2},\ldots$ is a sequence of positive numbers with limit zero.
I have a couple of questions:
- Any hints on how to prove the existence of these sequences?
- Does this have any relation with orthogonal projections?
- How can we define an orthogonal projection in a Hilbert space?
- Are they projecting to finite subspaces of $H$? If so, what are these finite subspaces?
- If $T$ is self-adjoint, would it be easier to prove that?