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, S is called bounded if for every neighborhood N of the zero vector there exists a scalar α such that

$S \subseteq \alpha N$

with

$\alpha N := \{ \alpha x \mid x \in N\}$.

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

$|\alpha| \ge t \Rightarrow S \subseteq \alpha N$.

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 (c_n) of scalars converging to 0 and all (similarly-indexed) countable subsets (x_n) of S, the sequence of their products (c_n x_n) 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

• 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

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 $L^p$ spaces for $0<p<1$ 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

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 0_M there exists a neighborhood w of 0_R such that w A ⊂ N.

See also

• Totally bounded space
• Local boundedness
• bounded function
• bounding point

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.