Definition:Compact Linear Transformation

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Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.

Let $T: X \to Y$ be a linear transformation.

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

Then $T$ is said to be a compact linear transformation, or simply compact if and only if $\map \cl {\map T {\operatorname {ball} X} }$ is compact in $Y$, where $\cl$ denotes topological closure.

Compact Operator

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

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

Also see

  • Results about compact linear transformations can be found here.