From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 12 Sep 2023 07:36:33 +0200
Subject: Initial commit

---
 .../the-fundamental-four/closed-graph-theorem.md   | 31 ++++++++++++++++++++++
 1 file changed, 31 insertions(+)
 create mode 100644 pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md

(limited to 'pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md')

diff --git a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
new file mode 100644
index 0000000..f8b8254
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+++ b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
@@ -0,0 +1,31 @@
+---
+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 %}
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