Balanced set

In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field K with an absolute value function | |) is a set S such that for all scalars α with |α| ≤ 1

$\alpha S \subseteq S$

where

$\alpha S := \{\alpha x \mid x \in S\} .$

The balanced hull or balanced envelope for a set S is the smallest balanced set containing S. It can be constructed as the intersection of all balanced sets containing S.

Examples

• The open and closed balls centered at 0 in a normed vector space are balanced sets.
• Any subspace of a real or complex vector space is a balanced set.
• The cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field K).
• Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do. As a result, ℂ and ℝ² are entirely different as far as their vector space structure is concerned.
• If p is a semi-norm on a linear space X, then for any constant c>0, the set {x ∈ X | p(x)≤c} is balanced.

Properties

• The union and intersection of balanced sets is a balanced set.
• The closure of a balanced set is balanced.
• By definition (not property), a set is absolutely convex if and only if it is convex and balanced

See also

• Star domain

References

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press. p. 4.
• W. Rudin (1990). Functional Analysis (2nd ed.). McGraw-Hill, Inc. ISBN 0-07-054236-8.
• H.H. Schaefer (1970). Topological Vector Spaces. GTM 3. Springer-Verlag. p. 11. ISBN 0-387-05380-8.