# Boundedness

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The term * bounded* appears in different parts of mathematics where a notion of "size" can be given. The basic intuitive meaning common to all of them is that something is of finite size, and that this is the case if it is smaller than some other object that has a finite size. (Otherwise it is

**unbounded**.)

- In topology, a subset of a metric space is
**bounded**if it can be contained in some ball of a certain radius. - In functional analysis, a subset
*A*of a topological vector space is**bounded**if for every neighbourhood*N*of the zero vector there exists some scalar α to that*A*is a subset of α*N* - A function is
**bounded**if its range is a bounded set. - A linear transformation is
**bounded**if the image of the unit ball is a bounded set. - A sequence is called
**bounded**if the set of its terms is bounded. - A partially ordered set is
**bounded**if it is has both a greatest element and a least element. - A function defined on an interval of the real line has
**bounded variation**if its graph has finite arclength. - In axiomatic set theory, the
**boundedness axiom**is a closely related and equivalent axiom to the axiom of replacement.