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 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 von Neumann and 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 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
* In a locally convex space the set of points of a Cauchy sequence is 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 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* 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 "o_M" there exists a neighborhood "w" of 0_"R" such that "w A ⊂ N".

See also

*Totally bounded
*Local boundedness
*bounded function

References

*
*