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 |.|) is a set S so 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[edit]

  • The open and closed balls 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.

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