---
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 $y=Tx$.