From 8857bb9fd92421b3aa2a3c1812345f34b6fa0b11 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Fri, 2 Oct 2026 15:38:20 -0400
Subject: [PATCH] Added counter examples to the open mapping theorem.

---
 src/fa/tvs/complete-metric.tex | 10 ++++++++++
 1 file changed, 10 insertions(+)

diff --git a/src/fa/tvs/complete-metric.tex b/src/fa/tvs/complete-metric.tex
index 6d5e0fc..8eefa3f 100644
--- a/src/fa/tvs/complete-metric.tex
+++ b/src/fa/tvs/complete-metric.tex
@@ -150,3 +150,13 @@
 
 \end{proof}
 
+
+\begin{example}
+    \label{example:open-mapping-counters}
+
+    Let $E = l^1(\natp; \complex)$ and $F$ be $l^1(\natp; \complex)$ equipped with the uniform norm, then the identity $E \to F$ is continuous but not open. 
+
+    Let $E = C^1([0, 1]; \complex)$ equipped with the uniform norm, $F = C([0, 1]; \complex)$, and $D: E \to F$ be the differentiation operator, then $D$ has closed graph, but is not continuous. 
+\end{example}
+
+