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Diffstat (limited to 'pages/distribution-theory/index.md')
| -rw-r--r-- | pages/distribution-theory/index.md | 29 |
1 files changed, 14 insertions, 15 deletions
diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md index b4b50a8..3055c8f 100644 --- a/pages/distribution-theory/index.md +++ b/pages/distribution-theory/index.md @@ -1,6 +1,6 @@ --- title: Distribution Theory -nav_order: 3 +nav_order: 5 has_children: true has_toc: false published: true @@ -10,17 +10,16 @@ published: true 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. +{% 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. +{% enddefinition %} |