From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 14 Feb 2024 07:24:38 +0100
Subject: Update

---
 .../functional-analysis-basics/reflexive-spaces.md | 86 +++++++++++-----------
 1 file changed, 44 insertions(+), 42 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 dee0e55..781fb1f 100644
--- a/pages/functional-analysis-basics/reflexive-spaces.md
+++ b/pages/functional-analysis-basics/reflexive-spaces.md
@@ -2,59 +2,59 @@
 title: Reflexive Spaces
 parent: Functional Analysis Basics
 nav_order: 2
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (Canonical Embedding)
->
-> Let $X$ be a normed space.
-> The mapping
->
-> $$
-> C : X \longrightarrow X'', \quad x \mapsto g_x,
-> $$
->
-> where the functional $g_x$ on $X'$ is defined by
->
-> $$
-> g_x(f) = f(x) \quad \text{for $f \in X'$,} 
-> $$
->
-> is called the *canonical embedding* of $X$ into its bidual $X''$.
-
-{: .lemma }
-> The canonical embedding $C : X \to X''$ of a normed space into its bidual
-> is well-defined and an embedding of normed spaces.
+{% definition Canonical Embedding %}
+Let $X$ be a normed space.
+The mapping
+
+$$
+C : X \longrightarrow X'', \quad x \mapsto g_x,
+$$
+
+where the functional $g_x$ on $X'$ is defined by
+
+$$
+g_x(f) = f(x) \quad \text{for $f \in X'$,} 
+$$
+
+is called the *canonical embedding* of $X$ into its bidual $X''$.
+{% enddefinition %}
+
+{% lemma %}
+The canonical embedding $C : X \to X''$ of a normed space into its bidual
+is well-defined and an embedding of normed spaces.
+{% endlemma %}
 
 {% proof %}
 {% endproof %}
 
 In particular, $C$ is isometric, hence injective.
 
-{: .definition-title }
-> Definition (Reflexivity)
->
-> A normed space is said to be *reflexive*
-> if the canonical embedding into its bidual
-> is surjective.
+{% definition Reflexivity %}
+A normed space is said to be *reflexive*
+if the canonical embedding into its bidual
+is surjective.
+{% enddefinition %}
 
 If a normed space $X$ is reflexive,
 then $X$ is isomorphic with $X''$, its bidual.
 James gives a counterexample for the converse statement.
 
-{: .theorem }
-> If a normed space is reflexive,
-> then it is complete; hence a Banach space.
+{% theorem %}
+If a normed space is reflexive,
+then it is complete; hence a Banach space.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
-{: .theorem }
-> If a normed space $X$ is reflexive,
-> then the weak and weak$^*$ topologies on $X'$ agree.
+{% theorem %}
+If a normed space $X$ is reflexive,
+then the weak and weak$^*$ topologies on $X'$ agree.
+{% endtheorem %}
 
 {% proof %}
 By definition, the weak and weak$^*$ topologies on $X'$
@@ -65,9 +65,10 @@ Since $X$ is reflexive, those sets are equal.
 
 The converse is true as well. Proof: TODO
 
-{: .theorem }
-> If a normed space $X$ is reflexive,
-> then its dual $X'$ is reflexive.
+{% theorem %}
+If a normed space $X$ is reflexive,
+then its dual $X'$ is reflexive.
+{% endtheorem %}
 
 {% proof %}
 Since $X$ is reflexive,
@@ -115,9 +116,10 @@ This shows that $D$ is surjective, hence $X'$ is reflexive.
 In fact, we have shown more: $D = (C')^{-1}$.
 {% endproof %}
 
-{: .theorem }
-> Every finite-dimensional normed space is reflexive.
->
+{% theorem %}
+Every finite-dimensional normed space is reflexive.
+{% endtheorem %}
 
-{: .theorem }
-> Every Hilbert space is reflexive.
+{% theorem %}
+Every Hilbert space is reflexive.
+{% endtheorem %}
-- 
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