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 $|\cdot |$) is a set S such that for all scalars $\alpha$ with $|\alpha |\leqslant 1$

$\alpha S\subseteq S$

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 $\mathbb {C} ,$ the field of complex numbers, as a 1-dimensional vector space. The balanced sets are $\mathbb {C}$ itself, the empty set and the open and closed discs centered at zero. Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at the origin will do. As a result, $\mathbb {C}$ and $\mathbb {R} ^{2}$ 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

$\{x\in X\mid p(x)\leqslant c\}$

is balanced.

Properties

• The union and intersection of balanced sets is a balanced set.
• The closure of a balanced set is balanced.
• The union of $\{0\}$ and the interior of a balanced set is balanced.
• 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.