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.
Read more about this topic: Balanced Set
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)