\begin{definition}[Definition:Compact Linear Transformation/Normed Vector Space/Definition 1]
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''' {{iff}}:
:$\map \cl {\map T {\operatorname {ball} X} }$ is compact in $\struct {Y, \norm \cdot_Y}$
where $\cl$ denotes topological closure.
\end{definition}