 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 .) is a set S so that for all scalars α with α ≤ 1
with
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.
Contents
Examples
 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).
 Consider C, the field of complex numbers, as a 1dimensional 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 R^{2} are entirely different as far as their vector space structure is concerned.
 If p is a seminorm on a linear space X, then for any constant c>0, the set {p≤c} is balanced.
Properties
 The union and intersection of balanced sets is a balanced set.
 The closure of a balanced set is balanced.
 By definition, 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 ed.). McGrawHill, Inc. ISBN 0070542368.
 H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. SpringerVerlag. p. 11. ISBN 0387053808.
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