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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/functional-analysis-basics/compact-operators.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
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Diffstat (limited to 'pages/functional-analysis-basics/compact-operators.md')
-rw-r--r-- | pages/functional-analysis-basics/compact-operators.md | 57 |
1 files changed, 26 insertions, 31 deletions
diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md index b114c24..92e94ba 100644 --- a/pages/functional-analysis-basics/compact-operators.md +++ b/pages/functional-analysis-basics/compact-operators.md @@ -3,42 +3,37 @@ 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 $Y$. +{% 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-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 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-title } -> Every compact linear operator is bounded. +{% proposition %} +Every compact linear operator is bounded. +{% endproposition %} -{: .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 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-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. +{% 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 %} |