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.

1 Properties
2 Absolutely convex hull
3 References
4 See also

Properties

A set $C$ is absolutely convex if and only if for any points $x_1, \, x_2$ in $C$ and any numbers $\lambda_1, \, \lambda_2$ satisfying $|\lambda_1| %2B |\lambda_2| \leq 1$ the sum $\lambda_1 x_1 %2B \lambda_2 x_2$ 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

$\mbox{absconv} A = \left\{\sum_{i=1}^n\lambda_i x_i�: n \in \N, \, x_i \in A, \, \sum_{i=1}^n|\lambda_i| \leq 1 \right\}$ .

References

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

Absolutely convex set

Contents

Properties

Absolutely convex hull

References

See also