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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 %}