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diff --git a/pages/more-functional-analysis/fixed-point-theorems/index.md b/pages/more-functional-analysis/fixed-point-theorems/index.md
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+---
+title: Fixed-Point Theorems
+parent: More Functional Analysis
+nav_order: 1
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+{% theorem * Banach Fixed-Point Theorem %}
+test
+{% endtheorem %}
diff --git a/pages/more-functional-analysis/index.md b/pages/more-functional-analysis/index.md
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+---
+title: More Functional Analysis
+nav_order: 4
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
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
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+---
+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
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+---
+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
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+---
+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 %}
diff --git a/pages/more-functional-analysis/topological-vector-spaces/index.md b/pages/more-functional-analysis/topological-vector-spaces/index.md
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+---
+title: Topological Vector Spaces
+parent: More Functional Analysis
+nav_order: 1
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+Let $X$ be a set.
+A *property* of subsets of $X$ is a set $P \subset \mathcal{P}(X)$.
+We say that a subset $A \subset X$ has the property $P$, if $A \in P$.
+A property $P$ of subsets of $X$ is said to be *stable under arbitrary intersections*,
+if for every family $F$ of subsets of $X$ with property $P$,
+the intersection $\bigcap F$ has the property $P$.
+In other words, $P$ is stable under arbitrary intersections iff
+$\bigcap F \in P$ for every subset $F \subset P$.
+In this definition the family $F$ is allowed to be empty,
+hence $\bigcap \emptyset = X$ needs to have the property $P$.
+
+For example, in a topological space $X$ the property of being a closed subset of $X$
+is stable under arbitrary intersections.
+
+If $P$ is stable under arbitrary intersections,
+and $A$ is a subset of $X$, which may or may not have the property $P$,
+then we define the *$P$-hull* of $A$ to be
+the intersection of all supersets $B \supset A$
+having have the property $P$.
+By definition, the $P$-hull of $A$ has the property $P$.
+Moreover, it is the smallest superset of $A$ with property $P$
+in the following sense: If $B$ is any superset of $A$ with property $P$,
+then $B$ contains the $P$-hull of $A$.
+
+For example, the "closed"-hull of a subset $A$ of a topological space
+is the closure of $A$.
+
+There are the dual notions of being *stable under arbitrary unions*
+and *$P$-core* with obvious definitions.
+
+{% definition Convex, Balanced, Absolutely Convex %}
+Let $X$ be a vector space over the field $\KK$.
+A subset $A \subset X$ is said to be
+- *convex* if
+- *balanced* if
+- *absolutely convex* if
+{% enddefinition %}
+
+{% theorem %}
+These properties of subsets of $X$
+are stable under arbitrary intersections.
+Thus we obtain the notions of
+*convex hull* $\co A$,
+*balanced hull* $\bal A$, and
+*absolutely convex hull* $\aco A$.
+{% endtheorem %}
diff --git a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md
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+++ b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md
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+---
+title: Polar Topologies
+parent: Topological Vector Spaces
+grand_parent: More Functional Analysis
+nav_order: 1
+---
+
+# {{ page.title }}
+
+# Dual pairs of vector spaces
+
+Recall that a *bilinear form* on two vector spaces $V$ and $W$ over a field $\KK$
+is a mapping $b : V \times W \to \KK$ which is linear in each of its arguments,
+that is, which satisfies
+
+$$
+\begin{align*}
+b(v_1+v_2,w) &= b(v_1,w) + b(v_2,w) &
+b(v,w_1+w_2) &= b(v,w_1) + b(v,w_2) \\
+b(\lambda v, w) &= \lambda \, b(v,w) &
+b(v, \lambda w) &= \lambda \, b(v,w)
+\end{align*}
+$$
+
+for all vectors $v,v_1,v_2 \in V$, $w,w_1,w_2 \in W$ and all scalars $\lambda \in \KK$.
+
+We say that the bilinear form $b : V \times W \to \KK$ is *nondegenerate*, if it has the properties
+
+$$
+\begin{gather*}
+\forall v \in V : \qquad ( \forall w \in W : \angles{v,w} = 0 ) \implies v = 0 \\
+\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0 
+\end{gather*}
+$$
+
+If $V$ is a vector space over $\KK$,
+let us denote its *algebraic dual* by $V^*$.
+Given a bilinear form $V \times W \to \KK$, consider the mappings
+
+$$
+c : V \to W*, c(v)(w) = b(v,w)
+\tilde{c} : W \to V*, \tilde{c}(w)(v) = b(v,w)
+$$
+
+Then $b$ is nondegenerate if and only if
+both $c$ and $\tilde{c}$ are injective.
+
+
+{% definition Dual Pair %}
+A *dual pair* (or *dual system* or *duality*) $\angles{V,W}$ over a field $\KK$ is constituted by
+two vector spaces $V$ and $W$ over $\KK$
+and a nondegenerate bilinear form $\angles{\cdot,\cdot} : V \times W \to \KK$.
+{% enddefinition %}
+
+(We resist saying that a dual pair is a triple ...)
+
+{% definition Weak Topology %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$.
+We define the *weak topology on $X$*, denoted by $\sigma(X,Y)$, as
+the [initial topology](/pages/general-topology/universal-constructions.html#initial-topology) induced by the maps
+$\angles{\cdot,y} : X \to \KK$, where $y \in Y$.
+Similarly, the *weak topology on $Y$*, denoted by $\sigma(Y,X)$, is
+the initial topology induced by the maps
+$\angles{x,\cdot} : Y \to \KK$, where $x \in X$.
+{% enddefinition %}
+
+{% theorem Weak Topologies are Locally Convex %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$.
+TODO
+{% endtheorem %}
+
+## The Canonical Pairing
+
+TODO: Def & Theorem (weak rep)
+
+{% definition Polar Set %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces.
+The *polar* of a subset $A \subset X$ is the set
+
+$$
+A^{\circ} = \braces{y \in Y : \abs{\angles{x,y}} \le 1 \ \forall x \in A}.
+$$
+
+The *polar* of a subset $B \subset Y$ is the set
+
+$$
+B^{\circ} = \braces{x \in X : \abs{\angles{x,y}} \le 1 \ \forall y \in B}.
+$$
+{% enddefinition %}
+
+Some authors define the polar with the condition $\Re \angles{x,y} \le 1$
+instead of $\abs{\angles{x,y}} \le 1$ and call *absolute polar* what we call polar.
+Some authors write $B_{\circ}$ for $B^{\circ}$.
+
+Note that the *bipolar* $A^{\circ\circ} = (A^{\circ})^{\circ}$ is a subset of $X$.
+
+{% theorem * Bipolar Theorem %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces
+and $A \subset X$. Then
+
+$$
+A^{\circ\circ} = \overline{\aco(A)},
+$$
+
+where the closure is taken with respect to the weak topology on $X$, that is $\sigma(X,Y)$.
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}