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title: Distribution Theory
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published: true
---

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As usual, let $\mathcal{S}$ denote the space of Schwartz test functions on $\RR^n$.

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> 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
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> - 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.