---
title: Closed Graph Theorem
parent: The Fundamental Four
grand_parent: Functional Analysis Basics
nav_order: 4
# cspell:words
---

# {{ page.title }}

{: .theorem-title }
> {{ page.title }}
> {: #{{ page.title | slugify }} }
>
> An (everywhere-defined) linear operator between Banach spaces is bounded
> iff its graph is closed.

We prove a slightly more general version:

{: .theorem-title }
> {{ page.title }}
> {: #{{ page.title | slugify }}-variant }
>
> Let $X$ and $Y$ be Banach spaces
> and $T : \dom{T} \to Y$ a linear operator
> with domain $\dom{T}$ closed in $X$.
> Then $T$ is bounded if and only if
> its graph $\graph{T}$ is closed.

{% proof %}
{% endproof %}