Uniform convergence

In calculus the uniform convergence describes the property of a sequence of functions to converge with an independent from the function argument " speed " to a limit function. In contrast to pointwise convergence of the notion of uniform convergence, important properties of the functions such as continuity, differentiability and Riemann integrability allowed to transfer to the limit function.

• 7.1 Uniform convergence of complex functional consequences 7.1.1 definition
• 7.1.2 Chordal uniform convergence
• 7.1.3 properties

Definition

Given a sequence of functions

Which every natural number assigns a real-valued function, and a function. All and are defined on the same set of definitions. The sequence converges to uniform when

One considers here the absolute difference of and for all from the domain. The amount of these differences is either unlimited or has a least upper bound, a supremum. Uniform convergence of opposite means that this supremum exists for almost all and goes to zero as tends to infinity.

One can also define this situation differently: All designations are as above. Then converges uniformly to if and only if there exists one for all, such that for all and for all:

Example

The sequence of functions

Converges in their own domain for uniformly to

Comparison between uniform and pointwise convergence

The choice of when to uniform convergence depends only on. In contrast, in the case of point -wise convergence on both as well as. Formulated to both concepts of convergence using quantifiers, we see that they differ in the order of the "Introduction" of and and thus the dependence of the two variables (see the underlined ):

That is, for pointwise convergence there must be for each and for each natural number, such that for all.

From the uniform convergence follows the pointwise convergence, but not vice versa. For example, the sequence of functions converges defined by

But pointwise to the zero function for each is not uniformly convergent sequence.

Designation

For the uniform convergence of a sequence of functions which tends towards one of the following designations is used mostly

Or

Or

Uniform convergence at a point

A sequence of functions called in the point to be uniformly convergent if

If, instead of for all the validity of the inequality is required for at least one, then that means the uniform convergence evenly convergent sequences are also uniformly convergent. The uniform convergence does not imply pointwise convergence.

Be

• The class of uniformly convergent sequences of functions,
• The class of each point uniformly convergent sequences of functions and
• The class of each point pointwise convergent sequences of functions.

Because of these assumptions.

The above-mentioned function sequence is within, so it is uniformly convergent, but not global in any point.

An example of a sequence of function is defined by

The sequence of functions converges pointwise to the zero function. For every rational number is in all of which is equal to or greater than the denominator in the fully shortened representation of the fraction. On the other hand, are on average a and an arbitrary interval only finitely many rational numbers. Therefore, for every and each number always ( infinitely many rational ) numbers whose distance is to be arbitrarily small and not related to. So the sequence converges uniformly at any point.

Conclusions

As already mentioned, the notion of uniform convergence allows starting from properties of the result statements about the limit function, which is not possible with pointwise convergence. The following are the designations are as in the definition above, is a real interval. This results in the following sentences:

Continuity

• It is a sequence of continuous functions. If uniformly converges to, then is continuous. Instead of calling for uniform convergence, it is sufficient to start with easy - uniform convergence.

• Let be a convergent sequence of functions from pointwise. All are still in this constantly. is in continuous if and only if the point is in uniform convergent.

• The amount of points uniform convergence as well as the set of points of uniform convergence of all pointwise convergent sequence of functions is Gδ - set.

• The uniformly convergent sequences of functions with compact domain are all equicontinuous.

• Let be a compact interval and a steady equi- on sequence. If converges pointwise to, then they also converges uniformly.

• Let be a sequence of functions with compact domain. if and only has a uniformly convergent subsequence, if equi is continuous and at each point of limited (Theorem of Arzelà - Ascoli ).

Differentiability

Integrability

For the Riemann integral on intervals integration and thresholding can be interchanged with uniform convergence:

An example of a pointwise, but not uniformly convergent sequence of functions, in which the integral can not be interchanged with the limit value, this function returns a result: for each function is defined by

Continuous and therefore Riemann integrable. For the integral

The sequence of functions converges pointwise to the zero function for all. is therefore

Pointwise convergence is therefore not sufficient, so limit and integral sign may be reversed.

Set of Dini

When a compact interval and a monotonic sequence of continuous functions (i.e., ≥ or ≤ for each and any ), which converges towards a point by point is also a continuous function, then converges uniformly.

Generalizations

Uniform convergence of complex functional consequences

Definition

The uniform convergence for complex sequences of functions is as in the case of real sequences of functions defined exactly. A sequence of functions

Is, against

Uniformly convergent if

Chordal uniform convergence

Is called chordal uniformly convergent if

In which

The name for chordal distance.

Be

• The class of convergent sequences of functions on evenly,
• The class of chordal on uniformly convergent sequences of functions and
• On the class of object a bounded function pointwise convergent sequences of functions.

It is

Properties

Similar to the uniform convergence of real sequences of functions of uniform limit can be interchanged with the differential or the integral curve in the complexes.

Uniform convergence in metric spaces

Be a lot, a metric space and a sequence of functions. This sequence of functions is called uniformly to convergent if one exists for all, so that

Applies.

Uniform convergence in uniform spaces

Entirely analogously can be uniform convergence for functions in a uniform space with a system of neighborhoods define: A filter (or more generally a filter base ) converges to the set of functions for a lot just then to a function, if there exists one for each neighborhood, so