pages/functional-analysis-basics/compact-operators.md


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

# {{ page.title }}

{% definition Compact Linear 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 $Y$.
{% enddefinition %}

{% proposition Characterization 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 subsequence.
{% endproposition %}

{% proposition %}
Every compact linear operator is bounded.
{% endproposition %}

{% proposition Compactness of Zero and Identity %}
The zero operator on any normed space is compact.
The identity operator on a normed space $X$ is compact if and only if $X$ has finite dimension.
{% endproposition %}

{% 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.
{% endproposition %}