In mathematics, a set is called bounded, if it is, in a certain sense, of finite size. If a set is not bounded, it is called unbounded. Two different definitions exists, one for metric spaces and standard calculus and one for topological vector spaces. For topological vector space with a metric the definitions do not coincide so it is necessary to infer from the context which definition is used.

Metric spaces

Simple definition

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.

General definition

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.

Topological vector spaces

A set in a topological vector space (X,τ) is called bounded or von Neumann bounded if every neighborhood of the zero vector can be inflated to include the set.

More formally given a topological vector space over a field F, S is called bounded if for every neighborhood of the zero vector N there exists a scalar α so that

with

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.

Topological modules

A subset A of a topological module M over a topological ring R is bounded iff for any neighborhood N of o_M there exists a neighborhood w of 0_R such that w A ⊂ N.

See also

• Totally bounded
• Locally bounded
• bounded function