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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

## Examples

## Locally bounded family

## Examples

_{0} is a real number, one can choose the neighborhood A to be the interval (x_{0}-1, x_{0}+1). Then for all x in this interval and for all n≥1 one has
_{0}|+1._{0} one can choose the neighborhood A to be R itself. Then we have
_{0} or its neighborhood A. This family is then more than locally bounded, it is actually uniformly bounded._{0} the values f_{n}(x_{0}) cannot be bounded as n 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.
### Locally bounded topological vector spaces

Let X be a topological vector space. Then a subset B ⊂ X is bounded if, for each open neighborhood U of 0 in X, there exists a number m > 0 such that
### Locally bounded functions

Let X be a topological space, Y a topological vector space, and f : X → Y a function. Then f is locally bounded if each point of X has a neighborhood whose image under f is bounded.## External links

A function f defined on some topological space X with real or complex values is called locally bounded,
if for any x_{0} in X there exists a neighborhood A of x_{0} such that
f (A) is a bounded set, that is, for some number M>0 one has

- $|f(x)|le\; M$

That is to say, for each x, one can find a constant depending on x, which is larger than the values of the function around x. Compare this with a bounded function, for which the constant does not depend on x. Obviously, if a function is bounded, then the function is locally bounded.

This definition can be extended to the case when f takes values in some metric space. Then, the inequality above needs to be replaced with

- $dleft(f(x),\; aright)le\; M$

- The function f: R → R

- $f(x)=frac\{1\}\{x^2+1\},$

- The function f: R → R

- $f(x)=2x+3,$

- The function f:R → R defined by

- $f(x)=frac\{1\}\{x\},$

A set (also called a family) U of functions defined on some topological space X with real or complex values is called locally bounded, if for any x_{0} in X there exists a neighborhood A of x_{0} and a positive number M such that

- $|f(x)|le\; M$

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.

- The family of functions f
_{n}:R→R

- $f\_n(x)=frac\{x\}\{n\}$

- $|f\_n(x)|le\; M$

- The family of functions f
_{n}:R→R

- $f\_n(x)=frac\{1\}\{x^2+n^2\}$

- $|f\_n(x)|le\; M$

- The family of functions f
_{n}:R→R

- $f\_n(x)=x+n$

- B ⊂ xU for all x > m.

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

- Theorem. A topological vector space X is locally bounded if, and only if, the identity mapping 1 : X → X is locally bounded.

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Last updated on Friday June 30, 2006 at 07:37:30 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday June 30, 2006 at 07:37:30 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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