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

---
 .../alaoglu-bourbaki-theorem.md                    | 28 ++++++++++++++++++++++
 .../locally-convex-spaces/index.md                 |  9 +++++++
 .../locally-convex-spaces/krein-milman-theorem.md  | 19 +++++++++++++++
 3 files changed, 56 insertions(+)
 create mode 100644 pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md
 create mode 100644 pages/more-functional-analysis/locally-convex-spaces/index.md
 create mode 100644 pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md

(limited to 'pages/more-functional-analysis/locally-convex-spaces')

diff --git a/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md
new file mode 100644
index 0000000..be424c3
--- /dev/null
+++ b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md
@@ -0,0 +1,28 @@
+---
+title: Alaoglu–Bourbaki Theorem
+parent: Locally Convex Spaces
+grand_parent: More Functional Analysis
+nav_order: 1
+---
+
+# {{ page.title }}
+
+Let $X$ be locally convex space and
+let $U \subset X$ be a neighborhood of zero.
+Let $X'$ denote the continuous dual of $X$.
+Recall that there is a canonical pairing
+
+$$
+X \times X' \to \CC, \quad (x,f) \mapsto \angles{x,f} = f(x).
+$$
+
+The weak topology on $X'$ with respect to this pairing
+is called weak\* topology.
+It is the weakest topology on $X'$ such that
+all evaluation maps $\angles{x,\cdot} : X \to \CC$ are continuous.
+The polar of $U$ is the subset $U^{\circ} \subset X'$.
+The theorem asserts that $U^{\circ}$ is compact in the weak\* topology.
+
+{% theorem * Alaoglu–Bourbaki Theorem %}
+The polar of a neighborhood of zero in a locally convex space is weak\* compact.
+{% endtheorem %}
diff --git a/pages/more-functional-analysis/locally-convex-spaces/index.md b/pages/more-functional-analysis/locally-convex-spaces/index.md
new file mode 100644
index 0000000..990ec99
--- /dev/null
+++ b/pages/more-functional-analysis/locally-convex-spaces/index.md
@@ -0,0 +1,9 @@
+---
+title: Locally Convex Spaces
+parent: More Functional Analysis
+nav_order: 2
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md b/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md
new file mode 100644
index 0000000..58b9182
--- /dev/null
+++ b/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md
@@ -0,0 +1,19 @@
+---
+title: Krein–Milman Theorem
+parent: Locally Convex Spaces
+grand_parent: More Functional Analysis
+nav_order: 2
+---
+
+# {{ page.title }}
+
+## Extreme Points
+
+{% definition Extreme Point %}
+Suppose $C$ is a convex subset of a vector space $X$.
+We say that an element $x \in C$ is an *extreme point* of $C$
+if
+{% enddefinition %}
+
+{% proof %}
+{% endproof %}
-- 
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