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diff --git a/pages/unbounded-operators/graph-and-closedness.md b/pages/unbounded-operators/graph-and-closedness.md new file mode 100644 index 0000000..a9bf738 --- /dev/null +++ b/pages/unbounded-operators/graph-and-closedness.md @@ -0,0 +1,22 @@ +--- +title: Graph and Closedness +parent: Unbounded Operators +nav_order: 1 +description: > + The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric + operator on a Hilbert space is bounded. We give a proof using the Uniform + Boundedness Theorem. We give another proof using the Closed Graph Theorem. +# spellchecker:dictionaries latex +# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz +--- + +# {{ page.title }} + + +{: .definition-title } + +> Definition (Graph of an Operator) +> +> The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$ +> is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$ +> where $x$ lies in the domain of $T$ and $y=Tx$. |