# 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. 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 *(c _{n})* of scalars converging to

*0*and all (similarly-indexed) countable subsets

*(x*of

_{n})*S*, the sequence of their products

*(c*necessarily converges to zero in

_{n}x_{n})*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 *0 _{M}* there exists a neighborhood

*w*of 0

_{R}such that

*w A ⊂ N*.

## Notes

- 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.