# Category:Compact Linear Transformations

This category contains results about **Compact Linear Transformations**.

Definitions specific to this category can be found in Definitions/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 $\mathsf{Pr} \infty \mathsf{fWiki}$ notationYou 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 $\struct {Y, \norm \cdot_Y}$.

## Pages in category "Compact Linear Transformations"

The following 13 pages are in this category, out of 13 total.