locally bounded

Suppose that X is a topological spaceMathworldPlanetmath and Y a metric space.


A set of functions f:XY is said to be locally bounded if for every xX, there exists a neighbourhood N of x such that is uniformly bounded on N.

In the special case of functions on the complex planeMathworldPlanetmath where it is often used, the definition can be given as follows.


A set of functions f:G is said to be locally bounded if for every aG there exist constants δ>0 and M>0 such that for all zG such that |z-a|<δ, |f(z)|<M for all f.

As an example we can look at the set of entire functionsMathworldPlanetmath where f(z)=z2+t for any t[0,1]. Obviously each such f is unboundedPlanetmathPlanetmath itself, however if we take a small neighbourhood around any point we can bound all f. Say on an open ball B(z0,1) we can show by triangle inequalityMathworldMathworldPlanetmathPlanetmath that |f(z)|(|z0|+1)2+1 for all zB(z0,1). So this set of functions is locally bounded.

Another example would be say the set of all analytic functionsMathworldPlanetmath from some region G to the unit discPlanetmathPlanetmath. All those functions are boundedPlanetmathPlanetmathPlanetmath by 1, and so we have a uniform bound even over all of G.

As a counterexample suppose the we take the constant functions fn(z)=n for all natural numbersMathworldPlanetmath n. While each of these functions is itself bounded, we can never find a uniform bound for all such functions.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title locally bounded
Canonical name LocallyBounded
Date of creation 2013-03-22 14:17:47
Last modified on 2013-03-22 14:17:47
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 9
Author jirka (4157)
Entry type Definition
Classification msc 30A99
Classification msc 54-00