Category:Definitions/Compact Linear Transformations
This category contains definitions related to Compact Linear Transformations.
Related results can be found in Category:Compact Linear Transformations.
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:
- $\map \cl {\map T {\operatorname {ball} X} }$ 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 $\struct {Y, \norm \cdot_Y}$.
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 $\struct {X, \norm \cdot_X}$.
Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: Use $\mathsf{Pr} \infty \mathsf{fWiki}$ notation You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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 $\struct {Y, \norm \cdot_Y}$.
Subcategories
This category has only the following subcategory.
C
Pages in category "Definitions/Compact Linear Transformations"
The following 10 pages are in this category, out of 10 total.
C
- Definition:Compact Linear Operator
- Definition:Compact Linear Transformation
- Definition:Compact Linear Transformation on Inner Product Space
- Definition:Compact Linear Transformation on Normed Vector Space
- Definition:Compact Linear Transformation/Inner Product Space
- Definition:Compact Linear Transformation/Inner Product Space/Definition 1
- Definition:Compact Linear Transformation/Inner Product Space/Definition 2
- Definition:Compact Linear Transformation/Normed Vector Space
- Definition:Compact Linear Transformation/Normed Vector Space/Definition 1
- Definition:Compact Linear Transformation/Normed Vector Space/Definition 2