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 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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[edit] 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.

[edit] 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

[edit] 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

[edit] 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.

[edit] See also

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