Topological Vector Spaces

Let be a set. A property of subsets of is a set . We say that a subset has the property , if . A property of subsets of is said to be stable under arbitrary intersections, if for every family of subsets of with property , the intersection has the property . In other words, is stable under arbitrary intersections iff for every subset . In this definition the family is allowed to be empty, hence needs to have the property .

For example, in a topological space the property of being a closed subset of is stable under arbitrary intersections.

If is stable under arbitrary intersections, and is a subset of , which may or may not have the property , then we define the -hull of to be the intersection of all supersets having have the property . By definition, the -hull of has the property . Moreover, it is the smallest superset of with property in the following sense: If is any superset of with property , then contains the -hull of .

For example, the “closed”-hull of a subset of a topological space is the closure of .

There are the dual notions of being stable under arbitrary unions and -core with obvious definitions.

Definition (Convex, Balanced, Absolutely Convex)

Let be a vector space over the field . A subset is said to be

  • convex if
  • balanced if
  • absolutely convex if

These properties of subsets of are stable under arbitrary intersections. Thus, we obtain the notions of convex hull , balanced hull , and absolutely convex hull .