Definition:Compact Linear Transformation/Normed Vector Space/Definition 1

Definition
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 in $\struct {X, \norm \cdot_X}$.

We say that $T$ is a compact linear transformation :


 * $\map \cl {\map T {\operatorname {ball} X} }$ is compact in $\struct {Y, \norm \cdot_Y}$

where $\cl$ denotes topological closure.

Also see

 * Equivalence of Definitions of Compact Linear Transformation on Normed Vector Space