# 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 \}.}$