Update

7 files changed, 58 insertions, 7 deletions

diff --git a/pages/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
index 0913776..1b6ff81 100644
--- a/pages/functional-analysis-basics/banach-alaoglu-theorem.md
+++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
@@ -1,7 +1,7 @@
 ---
 title: Banach–Alaoglu Theorem
 parent: Functional Analysis Basics
-nav_order: 3
+nav_order: 5
 ---
 
 # {{ page.title }}
diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md
index 92e94ba..dc20bad 100644
--- a/pages/functional-analysis-basics/compact-operators.md
+++ b/pages/functional-analysis-basics/compact-operators.md
@@ -1,7 +1,7 @@
 ---
 title: Compact Operators
 parent: Functional Analysis Basics
-nav_order: 4
+nav_order: 6
 published: false
 ---
 
diff --git a/pages/functional-analysis-basics/hilbert-spaces.md b/pages/functional-analysis-basics/hilbert-spaces.md
index b3ef52b..a77e5c7 100644
--- a/pages/functional-analysis-basics/hilbert-spaces.md
+++ b/pages/functional-analysis-basics/hilbert-spaces.md
@@ -1,7 +1,8 @@
 ---
 title: Hilbert Spaces
 parent: Functional Analysis Basics
-nav_order: 1
+nav_order: 7
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/functional-analysis-basics/normed-spaces/index.md b/pages/functional-analysis-basics/normed-spaces/index.md
new file mode 100644
index 0000000..c92d8c1
--- /dev/null
+++ b/pages/functional-analysis-basics/normed-spaces/index.md
@@ -0,0 +1,9 @@
+---
+title: Normed Spaces
+parent: Functional Analysis Basics
+nav_order: 1
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
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 %}
diff --git a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
index 1140e45..c88608b 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
@@ -3,6 +3,10 @@ title: Uniform Boundedness Theorem
 parent: The Fundamental Four
 grand_parent: Functional Analysis Basics
 nav_order: 2
+description: >
+  The Uniform Boundedness Theorem states that a pointwise bounded collection of
+  bounded linear operators from a Banach space into a normed space must be
+  uniformly bounded. We give a proof based on Baire’s Category Theorem.
 ---
 
 # {{ page.title }}
@@ -84,7 +88,9 @@ $$
 
 If $X$ is not complete, this may be false.
 
+{% comment %}
 TODO:
 - strong operator convergence
 - Kreyszig 4.9-5
 - Haase 15.6
+{% endcomment %}
diff --git a/pages/functional-analysis-basics/weak-convergence.md b/pages/functional-analysis-basics/weak-convergence.md
new file mode 100644
index 0000000..6e7f417
--- /dev/null
+++ b/pages/functional-analysis-basics/weak-convergence.md
@@ -0,0 +1,13 @@
+---
+title: Weak Convergence
+parent: Functional Analysis Basics
+nav_order: 3
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}