In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number.

Locally bounded function

A real****-valued**** or**** complex-valued**** function**** f defined on**** some**** topological space X is**** called**** a **** if**** for any x0 *in*** X there exists**** a neighborhood**** A of**** x0 such**** that**** f(A)**** is**** a bounded set.**** That is, for some number M > 0 one has In other words, for each x one can find a constant, depending on x, which is larger than all the values of the function in the neighborhood of x. Compare this with a bounded function, for which the constant does not depend on x. Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below). This definition can be extended to the case when f : X \to Y takes values in some metric space (Y, d). Then the inequality above needs to be replaced with where y \in Y is some point in the metric space. The choice of y does not affect the definition; choosing a different y will at most increase the constant r for which this inequality is true.

Examples

Locally bounded family

A set (also called a family) U of real-valued or complex-valued functions defined on some topological space X is called locally bounded if for any x_0 \in X there exists a neighborhood A of x_0 and a positive number M > 0 such that for all x \in A and f \in U. 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

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 (TVS) X is called bounded if for each neighborhood U of the origin in X there exists a real number s > 0 such that A **** is**** a TVS that**** possesses a bounded neighborhood**** of**** the origin****.**** By 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 pseudometrizable.

Locally bounded functions

Let f : X \to Y a function between topological vector spaces is said to be a locally bounded function if every point of X has a neighborhood whose image under f is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces: