Absolutely convex set

A set C in a real or complex vector space is said to be absolutely convex if it is convex and balanced.

Properties

A set C is absolutely convex if and only if for any points in C and any numbers satisfying the sum λ₁x₁ + λ₂x₂ belongs to C.

Since the intersection of any collection of absolutely convex sets is absolutely convex then for any subset A of a vector space one can define its absolutely convex hull to be the intersection of all absolutely convex sets containing A.

Absolutely convex hull

The absolutely convex hull of the set A assumes the following representation

.

References

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 4–6. 

See also

The Wikibook Algebra has a page on the topic of Vector spaces

• vector (geometric), for vectors in physics
• Vector field