Balanced Set - Examples

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 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.