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 |.|) is a set S so that for all scalars α with |α| ≤ 1

$\backslash alpha\; S\; \backslash subseteq\; S$

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

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 union of any family of balanced sets is a balanced set.
• The intersection of any family of balanced sets is a balanced set.
• The unit ball in a normed vector space is a balanced set.
• 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).
• If X=C and K=C, that means the vector space is the set of complex numbers (equipped with the usual operations) over the complex scalar field, the balanced sets are C itself, the empty set and the open and closed discs centered at 0. Contrariwise, if X=R² and K=R 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.

Rudin W. Functional Analysis, 2nd ed. McGraw-Hill,Inc.

Properties

• The union and intersection of balanced sets is a balanced set.
• By definition, a set is absolutely convex if and only if it is convex and balanced.

Linear algebra