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+---
+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.