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.

1 Locally bounded function
2 Examples
3 Locally bounded family
4 Examples
5 Topological vector spaces
- 5.1 Locally bounded topological vector spaces
- 5.2 Locally bounded functions
6 External links

[edit] Locally bounded function

A function f defined on some topological space X with real or complex values is called locally bounded, 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

$|f(x)|\le M$

for all x in A.

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

$d\left(f(x), a\right)\le M$

for all x in A, where d is the distance function in the metric space, and a is some point in the metric space. The choice of a does not affect the definition. Choosing a different a will at most increase the constant M for which this inequality is true.

[edit] Examples

The function f: R → R

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

is bounded, because 0≤ f (x) ≤ 1 for all x. Therefore, it is also locally bounded.

The function f: R → R

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

is not bounded, as it becomes extremely large when x is large. However, it is locally bounded.

The function f:R → R defined by

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

for x ≠ 0 and taking the value 0 for x=0 is not locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.

[edit] Locally bounded family

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 x₀ in X there exists a neighborhood A of x₀ and a positive number M such that

$|f(x)|\le M$

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.

[edit] Examples

The family of functions f_n:R→R

$f_n(x)=\frac{x}{n}$

where n = 1, 2, ... is uniformly bounded. Indeed, if x₀ is a real number, one can choose the neighborhood A to be the interval (x₀-1, x₀+1). Then for all x in this interval and for all n≥1 one has

$|f_n(x)|\le M$

with M=|x₀|+1.

The family of functions f_n:R→R

$f_n(x)=\frac{1}{x^2+n^2}$

is locally bounded. For any x₀ one can choose the neighborhood A to be R itself. Then we have

$|f_n(x)|\le M$

with M=1. Note that the value of M does not depend on the choice of x₀ or its neighborhood A. This family is then more than locally bounded, it is actually uniformly bounded.

The family of functions f_n:R→R

f n (x) = x + n

is not locally bounded. Indeed, for any x₀ the values f_n(x₀) cannot be bounded as n tends toward infinity.

[edit] 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.

[edit] 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

B ⊂ xU for all x > m.

A topological vector space is said to be locally bounded if X admits a bounded open neighborhood of 0.

[edit] 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.

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.