Definition:Compact Linear Transformation
Normed Vector Space
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:
- $\map \cl {T \sqbrk {\operatorname {ball} X} }$ 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 $\struct {Y, \norm \cdot_Y}$.
Inner Product Space
Definition 1
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.
Definition 2
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 $T : X \to Y$ be a linear transformation.
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 known as
If $T$ is a compact linear transformation, we often simply say that $T$ is compact.
Also see
- Results about compact linear transformations can be found here.