functions defined on some topological space is called locally bounded if for any there exists a neighborhood of and a positive number such that for all and In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant. This definition can also be extended to the case when the functions in the family U take values in some metric space, by again replacing the absolute value with the distance function. Examples The family of functions where is locally bounded. Indeed, if is a real number, one can choose the neighborhood to be the interval Then for all in this interval and for all one has with Moreover, the family is uniformly bounded, because neither the neighborhood nor the constant depend on the index The family of functions is locally bounded, if is greater than zero. For any one can choose the neighborhood to be itself. Then we have with Note that the value of does not depend on the choice of x0 or its neighborhood This family is then not only locally bounded, it is also uniformly bounded. The family of functions is locally bounded. Indeed, for any the values cannot be bounded as tends toward infinity. Topological vector spaces Local boundedness may also refer to a property of topological vector spaces, or of functions from a topological space into a topological vector space (TVS). Locally bounded topological vector spaces A subset of a topological vector space is call is bounded if for each neighborhood of the origin in there exists a real number such that A is a topological vector space that posses a bounded neighborhood of the origin. By the Kolmogorov's normability criterion, this is true of a locally convex space if and only if the topology of the TVS is induced by some seminorm. In particular, every locally bounded TVS is locally convex and pseudometrizable. Locally bounded functions Let a function between topological vector spaces is said to be a locally bounded function if every point of has a neighborhood whose image under is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces: Theorem. A topological vector space is locally bounded if and only if the identity map is locally bounded. See also External links PlanetMath entry for Locally Bounded nLab entry for Locally