\begin{definition}[Definition:Compact Linear Transformation]
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''' {{iff}} $\map \cl {\map T {\operatorname {ball} X} }$ is compact in $Y$, where $\cl$ denotes topological closure.
\end{definition}