Absolutely convex set

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

Definition

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

Relationship with seminorms

If U is an absolutely convex neighborhood of the topological vector space E, then defines a semi-norm on E. It is

It is also called the Minkowski functional on U.

Easy, one can show that every locally convex vector space has a base of neighborhoods of absolutely convex sets. Using the Minkowski functionals can also be described by the semi-norms topology so. This clarifies the relationship between the two in the article about locally convex spaces given definitions.

Absolutely convex cover of

Since averages of absolutely convex sets are obviously absolutely convex again, lots of 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

Source

  • R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg, 1992 ISBN 978-3-528-07262-9
  • Functional Analysis
Balanced set Base (topology)
25431
de