pages/unbounded-operators/graph-and-closedness.md


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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.
---

# {{ page.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 $y=Tx$.
{% enddefinition %}