Boundedness
From Exampleproblems
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 αN
- 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.
