Harman Patil (Editor)

Balanced set

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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

Contents

α S ⊆ S

where

α S := { α x ∣ x ∈ 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.
  • A set is absolutely convex if and only if it is convex and balanced

    • References

    Balanced set Wikipedia
    (Text) CC BY-SA