---
title: Compact Operators
parent: Functional Analysis Basics
nav_order: 4
published: false
# cspell:words
---

# {{ page.title }}

{: .definition-title }
> Definition (Compact Linear Operator)
> {: #compact-operator }
>
> A linear operator $T : X \to Y$,
> where $X$ and $Y$ are normed spaces,
> is said to be a  *compact linear operator*,
> if for every bounded subset $M \subset X$
> the image $TM$ is relatively compact in $Y$.

{: .proposition-title }
> Proposition (Characterisation of Compactness)
>
> Let $X$ and $Y$ be normed spaces.
> A linear operator $T : X \to Y$ is compact if and only if
> for every bounded sequence $(x_n)$ in $X$
> the image sequence $(Tx_n)$ in $Y$ has a convergent subseqence.

{: .proposition-title }
> Every compact linear operator is bounded.

{: .proposition-title }
> Proposition (Compactness of Zero and Identity)
>
> The zero operator on any normed space is compact.
> The indentity operator on a normed space $X$ is compact if and only if $X$ has finite dimension.

{: .proposition-title }
> Proposition (The Space of Compact Linear Operators)
>
> The set $C(X,Y)$ of compact linear operator from a normed space $X$ into a normed space $Y$
> form a linear subspace of the space $B(X,Y)$ of bounded linear operators from $X$ into $Y$.
> If $Y$ is a Banach space, then $C(X,Y)$ is a closed linear subspace of the Banach space
> $B(X,Y)$ and hence itself a Banach space.