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.) ${\displaystyle \lambda ,\mu \in {\mathbb {K} }}$ ${\displaystyle |\lambda |+|\mu |\leq 1}$ ${\displaystyle x,y\in A}$ ${\displaystyle \lambda x+\mu y\in A}$ ${\displaystyle {\mathbb {K} }}$

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 ${\displaystyle p_{U}(x):=\inf \left\{t>0\,\left|\,x\in tU\right.\right\}}$

${\displaystyle U^{\circ }=\left\{x\in E\,\left|\,p_{U}(x)<1\right.\right\}\subset U\subset \left\{x\in E\,\left|\,p_{U}(x)\leq 1\right.\right\}={\overline {U}}}$.

It is called also the Minkowski functional to U . ${\displaystyle p_{U}}$

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 ${\displaystyle \Gamma M}$ ${\displaystyle \Gamma M=\left\{\sum _{j=1}^{n}\lambda _{j}x_{j}\,\left|\,\lambda _{j}\in {\mathbb {K} },\,\sum _{j=1}^{n}\left|\lambda _{j}\right|\leq 1,\,x_{j}\in M,\,n\in {\mathbb {N} }\right.\right\}.}$

source

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