Closure of Range of Compact Linear Transformation is Separable
Theorem
Let $H, K$ be Hilbert spaces.
Let $T \in B_0 \left({H, K}\right)$ be a compact linear transformation.
Then $\operatorname{cl} \left({\operatorname{ran} T}\right)$ is separable.
Proof
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Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $II.4.5$
