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.
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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
This is equivalent to the condition that S is absorbed by every neighborhood of the zero vector, i.e., that for all neighborhoods N, there exists t such that
Bounded subsets of a topological vector space over the real or complex field can also be characterized by their sequences, for S is bounded in X if and only if for all sequences (cn) of scalars converging to 0 and all (similarly-indexed) countable subsets (xn) of S, the sequence of their products (cn xn) necessarily converges to zero in X.
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
Properties
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 0M there exists a neighborhood w of 0R such that w A ⊂ N.