From 8b9bb9346c217874670b0f1798ab6f1cb28fdb83 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 20 Feb 2024 12:01:07 +0100
Subject: Update

---
 .../functional-analysis-basics/reflexive-spaces.md | 30 +++++++++++++++++++---
 1 file changed, 26 insertions(+), 4 deletions(-)

(limited to 'pages/functional-analysis-basics/reflexive-spaces.md')

diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md
index 58ca1d3..69fefc4 100644
--- a/pages/functional-analysis-basics/reflexive-spaces.md
+++ b/pages/functional-analysis-basics/reflexive-spaces.md
@@ -1,7 +1,9 @@
 ---
 title: Reflexive Spaces
 parent: Functional Analysis Basics
-nav_order: 2
+nav_order: 4
+description: >
+  A normed space is said to be reflexive if the canonical embedding into its bidual is surjective.
 ---
 
 # {{ page.title }}
@@ -28,10 +30,30 @@ The canonical embedding $C : X \to X''$ of a normed space into its bidual
 is well-defined and an embedding of normed spaces.
 {% endlemma %}
 
+In particular, $C$ is isometric, hence injective.
+
 {% proof %}
-{% endproof %}
+We have to show that, for any given $x \in X$,
+$g_x$ is a bounded linear functional on $X'$.
+Linearity follows from the fact that
+the vector space structure on $X'$ is given by pointwise operations.
+To see that $g_x$ is bounded, observe that
 
-In particular, $C$ is isometric, hence injective.
+$$
+\abs{g_x(f)} = \abs{f(x)} \le \norm{f} \norm{x}
+$$
+
+holds for all $f \in X'$.
+Moreover, this implies that $\norm{g_x} \le \norm{x}$.
+Thanks to
+[Hahn–Banach](/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.html#hahn-banach-theorem-existence-of-functionals),
+we know that there exists a bounded linear functional
+$f \in X'$ with $\norm{f} = 1$ such that $f(x) = \norm{x}$;
+hence, $\norm{g_x} = \norm{x}$.
+This means that the mapping $x \mapsto g_x$ is isometric.
+Clearly, this mapping is also linear, and thus an embedding
+of normed spaces.
+{% endproof %}
 
 {% definition Reflexivity %}
 A normed space is said to be *reflexive*
@@ -40,7 +62,7 @@ is surjective.
 {% enddefinition %}
 
 If a normed space $X$ is reflexive,
-then $X$ is isomorphic with $X''$, its bidual.
+then $X$ is isometrically isomorphic with $X''$, its bidual.
 James gives a counterexample for the converse statement.
 
 {% theorem %}
-- 
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