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 with

where

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 the field of complex numbers, as a 1-dimensional vector space. The balanced sets are 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, and are entirely different as far as scalar multiplication is concerned.
  • If is a semi-norm on a linear space then for any constant the set
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 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

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.
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