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+---
+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.