Bounded set (topological vector space)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. Conversely a set that is not bounded is called unbounded.
Bounded sets are a natural way to define a locally convex polar topologies on the vector spaces in a dual pair, as the polar of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
In locally convex topological vector spaces the topology τ of the space can be specified by a family P of semi-norms. An equivalent characterization of bounded sets in this case is, a set S in (X,P) is bounded if and only if it is bounded for all semi normed spaces (X,p) with p a semi norm of P.
Examples and nonexamples
- Every finite set of points is bounded
- The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not to be bounded.
- Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topology the converse is also true.
- A (non null) subspace of a Hausdorff topological vector space is not bounded
- The closure of a bounded set is bounded.
- In a locally convex space, the convex envelope of a bounded set is bounded. (Without local convexity this is false, as the spaces for have no nontrivial open convex subsets.)
- The finite union or finite sum of bounded sets is bounded.
- Continuous linear mappings between topological vector spaces preserve boundedness.
- A locally convex space is seminormable if and only if there exists a bounded neighbourhood of zero.
- The polar of a bounded set is an absolutely convex and absorbing set.
- A set A is bounded if and only if every countable subset of A is bounded
The definition of bounded sets can be generalized to topological modules. A subset A of a topological module M over a topological ring R is bounded if for any neighborhood N of 0M there exists a neighborhood w of 0R such that w A ⊂ N.