Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces and $T: \mathcal{H}_1 \rightarrow \mathcal{H}_2$ a compact operator.
I want to show that $(\ker T)^\perp$ and $\text{ran}\ T$ are separable.
Since $T$ maps to a Hilbert space, there is a sequence $(T_n)_{n \in \mathbb{N}}$ of finite-dimensional operators that converges to $T$ in the operator norm.
Unfortunately, I was unable to make proper use of this fact. Or do I have to use another approach?
I don't know much about compact operators between Hilbert spaces…
Can someone help me to get started on this?
[Update on the Definitions I use here:]
A Hilbert space is an inner product space $(\mathcal{H}, \langle .,.\rangle)$ that is complete with respect to the norm that is induced by $\|x\|=\langle x,x\rangle^{1/2}$.
A linear space $\mathcal{X}$ is separable iff there exists a dense countable subset of $\mathcal{X}$, where countable means finite or countably infinite.
Best Answer
Here are some remarks to get you started.
With knowledge of these facts, you should be able to reduce one of your statements to the other. You might also consider trying to prove a lemma along the lines of