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author | Justin Gassner <justin.gassner@mailbox.org> | 2023-09-12 07:36:33 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-01-13 20:41:27 +0100 |
commit | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (patch) | |
tree | dc42d2ae9b4a8e7ee467f59e25c9e122e63f2e04 /pages/distribution-theory/index.md | |
download | site-777f9d3fd8caf56e6bc6999a4b05379307d0733f.tar.zst |
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diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md new file mode 100644 index 0000000..b4b50a8 --- /dev/null +++ b/pages/distribution-theory/index.md @@ -0,0 +1,26 @@ +--- +title: Distribution Theory +nav_order: 3 +has_children: true +has_toc: false +published: true +--- + +# {{ page.title }} + +As usual, let $\mathcal{S}$ denote the space of Schwartz test functions on $\RR^n$. + +{: .definition-title } +> Definition (Operator Valued Distribution) +> +> Let $\hilb{H}$ be a Hilbert space. +> An *operator valued tempered distribution* $\Phi$ (on $\RR^n$) +> is a mapping that associates to each test function $f \in \mathcal{S}$ +> an unbounded linear operator $\Phi(f)$ in $\hilb{H}$ such that +> {: .mb-0 } +> +> {: .my-0 } +> - there is a dense linear subspace $\mathcal{D}$ of $\hilb{H}$ that +> is contained in the domain of all the $\Phi(f)$ +> - for every fixed pair of vectors $\phi, \psi \in \hilb{D}$ +> the mapping $f \mapsto \innerp{\phi}{\Phi(f) \psi}$ is a tempered distribution. |