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+---
+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 $y=Tx$.