# Topological Vector Spaces

Let $X$ be a set. A *property* of subsets of $X$ is a set $P⊂P(X)$. We say that a subset $A⊂X$ has the property $P$, if $A∈P$. 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 $⋂F$ has the property $P$. In other words, $P$ is stable under arbitrary intersections iff $⋂F∈P$ for every subset $F⊂P$. In this definition the family $F$ is allowed to be empty, hence $⋂∅=X$ 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 $B⊃A$ 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 $K$. A subset $A⊂X$ is said to be

convexifbalancedifabsolutely convexif

These properties of subsets of $X$ are stable under arbitrary intersections. Thus, we obtain the notions of

convex hull$coA$,balanced hull$balA$, andabsolutely convex hull$acoA$.