bounded function

Definition Suppose X is a nonempty set. Then a function f:X is a if there exist a C< such that |f(x)|<C for all xX. The set of all bounded functions on X is usually denoted by B(X) ([1], pp. 61).

Under standard point-wise additionPlanetmathPlanetmath and point-wise multiplication by a scalar, B(X) is a complex vector space.

If fB(X), then the sup-norm, or uniform norm, of f is defined as


It is straightforward to check that |||| makes B(X) into a normed vector spacePlanetmathPlanetmath, i.e., to check that |||| satisfies the assumptionsPlanetmathPlanetmath for a norm.

0.0.1 Example

Suppose X is a compactPlanetmathPlanetmath topological spaceMathworldPlanetmath. Further, let C(X) be the set of continuousPlanetmathPlanetmath complex-valued functions on X (with the same vector spaceMathworldPlanetmath structureMathworldPlanetmath as B(X)). Then C(X) is a vector subspace of B(X).


  • 1 C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, 2nd ed., Academic Press, 1990.
Title bounded function
Canonical name BoundedFunction
Date of creation 2013-03-22 13:44:06
Last modified on 2013-03-22 13:44:06
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Definition
Classification msc 46-00
Defines supremum norm
Defines sup norm
Defines sup-norm
Defines uniform norm
Defines bounded function
Defines unbounded function