# Closure of Range of Compact Linear Transformation is Separable

From ProofWiki

## 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

## Sources

- John B. Conway:
*A Course in Functional Analysis*(1990)... (previous)... (next) $II.4.5$