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+---
+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 %}