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.
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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
- <math>S \subseteq \alpha N</math>
with
- <math>\alpha N := \{ \alpha x \mid x \in N\}</math>.
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 precompact 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<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 oM there exists a neighborhood w of 0R such that w A ⊂ N.
