# 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 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

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

${\displaystyle S\subseteq \alpha N}$

with

${\displaystyle \alpha N:=\{\alpha x\mid x\in N\}}$.

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
• 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

## 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.