Definition:Compact Linear Transformation/Inner Product Space/Definition 1
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Definition
Let $\struct {X, \innerprod \cdot \cdot_X}$ and $\struct {Y, \innerprod \cdot \cdot_Y}$ be inner product spaces.
Let $\norm \cdot_X$ and $\norm \cdot_Y$ be the inner product norms of $\struct {X, \innerprod \cdot \cdot_X}$ and $\struct {Y, \innerprod \cdot \cdot_Y}$ respectively.
Let $\operatorname {ball} X$ be the closed unit ball in $\struct {X, \norm \cdot_X}$.
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We say that $T$ is a compact linear transformation if and only if:
- $\map \cl {\map T {\operatorname {ball} X} }$ is compact in $\struct {Y, \norm \cdot_Y}$
where $\cl$ denotes topological closure.
Also see
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