# Local boundedness

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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[edit]

A real-valued or complex-valued function defined on some topological space is called a **locally bounded functional** if for any there exists a neighborhood of such that is a bounded set. That is, for some number one has

In other words, for each one can find a constant, depending on which is larger than all the values of the function in the neighborhood of Compare this with a bounded function, for which the constant does not depend on 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 takes values in some metric space Then the inequality above needs to be replaced with

## Examples[edit]

- The function defined by is bounded, because for all Therefore, it is also locally bounded.

- The function defined by is
*not*bounded, as it becomes arbitrarily large. However, it*is*locally bounded because for each in the neighborhood where

- The function defined by is neither bounded
*nor*locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.

- Any continuous function is locally bounded. Here is a proof for functions of a real variable. Let be continuous where and we will show that is locally bounded at for all Taking ε = 1 in the definition of continuity, there exists such that for all with . Now by the triangle inequality, which means that is locally bounded at (taking and the neighborhood ). This argument generalizes easily to when the domain of is any topological space.

- The converse of the above result is not true however; that is, a discontinuous function may be locally bounded. For example consider the function given by and for all Then is discontinuous at 0 but is locally bounded; it is locally constant apart from at zero, where we can take and the neighborhood for example.

## Locally bounded family[edit]

A set (also called a family) *U* of real-valued or complex-valued functions defined on some topological space is called **locally bounded** if for any there exists a neighborhood of and a positive number such that

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[edit]

- 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 haswith 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 havewith Note that the value of does not depend on the choice of x
_{0}or its neighborhood This family is then not only locally bounded, it is also uniformly bounded.

- The family of functions is
*not*locally bounded. Indeed, for any the values cannot be bounded as tends toward infinity.

## Topological vector spaces[edit]

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[edit]

A subset of a topological vector space (TVS) is called **bounded** if for each neighborhood of the origin in there exists a real number such that

**locally bounded TVS**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 locally convex and pseudometrizable.

### Locally bounded functions[edit]

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[edit]

- Bornological space – Space where bounded operators are continuous
- Bounded operator – Linear transformation between topological vector spaces
- Bounded set (topological vector space) – Generalization of boundedness