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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 $y=Tx$.
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