Bounded set (topological vector space)

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For bounded sets in general, see bounded set.

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[edit]

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

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[edit]

  • 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[edit]

  • 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[edit]

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 0M there exists a neighborhood w of 0R such that w A ⊂ N.

See also[edit]

References[edit]