# Bounded set (topological vector space)

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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

with

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 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

## 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

## References

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