# 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:

$\map \cl {\map T {\operatorname {ball} X} }$ is compact in $\struct {Y, \norm \cdot_Y}$

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 $\struct {X, \norm \cdot_X}$.

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.