bounded set

In mathematics, a set is called bounded, if it is, in a certain sense, of finite size. This definition will be made more specific below. If a set is not bounded, it is called unbounded.

Calculus

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.

Metric spaces

A subset S of 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. M is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Properties which are similar to boundedness but stronger, that is they imply boundedness, are total boundedness and compactness.

Functional analysis

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

Bounded Set

Calculus

Metric spaces

Functional analysis

See also