# Absolutely convex set

**Absolutely convex sets** play an important role in the theory of locally convex spaces , as they naturally lead to semi-norms .

## definition

A subset A of a real or complex vector space is called **absolutely convex** if for all with and all always holds. Thus A is absolutely convex if and only if A is balanced and convex . (Stands for the field of real or complex numbers.)

## Relationship to semi-norms

U is an absolutely zero neighborhood of the topological vector space *E* , defined as a semi-norm on *E* . It applies

- .

It is called also the Minkowski functional to *U* .

It is easy to show that every locally convex vector space has a zero neighborhood basis of absolutely convex sets. With the help of the Minkowski functionals, the topology can also be described by semi-norms. This clarifies the relationship between the two definitions given in the article on locally convex spaces .

## Absolutely convex hull

Since intersections of absolutely convex sets are obviously absolutely convex again, every set M of a real or complex vector space is contained in a smallest absolutely convex set. This is called the **absolutely convex hull** of M. It is true

## source

- R. Meise, D. Vogt:
*Introduction to Functional Analysis*, Vieweg, 1992 ISBN 978-3-528-07262-9