Definition:Compact Linear Transformation/Normed Vector Space
< Definition:Compact Linear Transformation(Redirected from Definition:Compact Linear Transformation on Normed Vector Space)
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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.
Definition 1
Let $\operatorname {ball} X$ be the closed unit ball in $\struct {X, \norm \cdot_X}$.
We say that $T$ is a compact linear transformation if and only if:
- $\map \cl {\map T {\operatorname {ball} X} }$ is compact in $\struct {Y, \norm \cdot_Y}$
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where $\cl$ denotes topological closure.
Definition 2
We say that $T$ is a compact linear transformation if and only if:
- for each bounded sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$:
- the sequence $\sequence {T x_n}_{n \mathop \in \N}$ has a subsequence convergent in $\struct {Y, \norm \cdot_Y}$.
Also see
- Equivalence of Definitions of Compact Linear Transformation on Normed Vector Space
- Definition:Compact Linear Operator on Normed Vector Space
Sources
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