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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.) For the precise definition no precise definition of 'size' is needed.

Table of contents
1 Calculus
2 Metric spaces
3 Functional analysis


A set S of real numbers is called bounded above if there is a real number k such that k > s for all s in S. The number k is called an upper bound of S. The terms bounded below and lower bound are similarly defined. A set S is bounded if it is bounded both above and below. Therefore, a set is bounded if it is contained in a finite interval.

A function f : X -> R is bounded on X if its image f(X) is a bounded subset of R.

Metric spaces

A set S in a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e. if there exists x in M and r > 0 such that for all s in S, we have d(x, s) < r.

Functional analysis

A set S in a topological vector space is bounded if it is contained in some multiple of every basic neighbourhood of zero. A bounded linear operator is continuous.