---
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 %}