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.

Definition

Given a topological vector space (X,τ) over a field F, a subset S of X is called bounded if for every neighborhood N of the zero vector there exists a scalar α such that

with

.

This is equivalent[1] to the condition that S is absorbed by every neighborhood of the zero vector, i.e., that for all neighborhoods N, there exists t such that

.

The collection of all bounded sets on a topological vector space X is called the bornology of X.

Bounded subsets of a topological vector space over the real or complex field can also be characterized by their sequences, for S is bounded in X if and only if for all sequences (cn) of scalars converging to 0 and all (similarly-indexed) countable subsets (xn) of S, the sequence of their products (cn xn) necessarily converges to zero in X.

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

  • In any topological vector space, finite sets are bounded, using that the origin has a local base of absorbent sets.
  • The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not 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.

Properties

  • 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 has a bounded neighborhood of zero if and only if its topology can be defined by a single seminorm.
  • 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

Generalization

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.

See also

Notes

  1. ↑ Schaefer 1970, p. 25.

References

  • Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 44–46.
  • H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 25–26. ISBN 0-387-05380-8.
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