Bounded set (topological vector space)

For bounded sets in general, see bounded set.

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 which 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.

1 Definition
2 Examples and nonexamples
3 Properties
4 Generalization
5 See also
6 References

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 so that

$S \subseteq \alpha N$

with

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

In other words a set is called bounded if it is absorbed by every neighborhood of the zero vector.

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.

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.