diff options
author | Justin Gassner <justin.gassner@mailbox.org> | 2023-09-12 07:36:33 +0200 |
---|---|---|
committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-01-13 20:41:27 +0100 |
commit | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (patch) | |
tree | dc42d2ae9b4a8e7ee467f59e25c9e122e63f2e04 /pages/functional-analysis-basics/compact-operators.md | |
download | site-777f9d3fd8caf56e6bc6999a4b05379307d0733f.tar.zst |
Initial commit
Diffstat (limited to 'pages/functional-analysis-basics/compact-operators.md')
-rw-r--r-- | pages/functional-analysis-basics/compact-operators.md | 44 |
1 files changed, 44 insertions, 0 deletions
diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md new file mode 100644 index 0000000..b114c24 --- /dev/null +++ b/pages/functional-analysis-basics/compact-operators.md @@ -0,0 +1,44 @@ +--- +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. |