Definition:Compact Linear Transformation

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Let $H, K$ be Hilbert spaces.

Let $T: H \to K$ be a linear transformation.

Let $\operatorname{ball} H$ be the closed unit ball of $H$.

Then $T$ is said to be a compact linear transformation, or simply compact if and only if $\operatorname{cl} \left({T \left({\operatorname{ball} H}\right) }\right)$ is compact in $K$, where $\operatorname{cl}$ denotes closure.

Compact Operator

When $H$ and $K$ are equal, one speaks about compact (linear) operators instead.

This is in line with the definition of a linear operator.

Also see