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Wikipedia / Local boundedness

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 defined on some topological space is called a there exists a neighborhood is a bounded set

Compare this with a bounded function Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below).

takes values in some metric space for which this inequality is true.

Examples

  • Therefore, it is also locally bounded.

    is bounded, as it becomes arbitrarily large. However, it

    is neither bounded locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.

    . Now by the triangle inequality is any topological space.

    for example.

Locally bounded family

A set (also called a family) U is called locally bounded there exists a neighborhood 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

  • Moreover, the family is uniformly bounded

    does not depend on the choice of x0 This family is then not only locally bounded, it is also uniformly bounded.

    is 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 is called bounded 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

a function between topological vector spaces is said to be a locally bounded function has a neighborhood whose image is bounded.

The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:

  1. Theorem. is locally bounded if and only if the identity map is locally bounded.

See also