The uniform convergence of a continuation of functions is a form of convergence more demanding than the simple Convergence. This last request indeed only that, for each point , the continuation has a limit. Convergence becomes uniform when all the continuations advances towards their respective limit with a kind of “overall movement”.
In the case of numerical functions of a variable, the concept takes a obviously geometrical form: the graph of the function “approaches” that of the limit.
Note:: the proposal is equivalent to:
One can ask for a posteriori which is the difference between the simple Convergence of a continuation in functions and the uniform convergence . Indeed, the continuation of functions converges simply towards on if:
In the case of the simple convergence , for any element , one can find a row from which the distance becomes very small. A priori , if one chooses a other than X then the row from which the distance becomes very small will be different.
- In the case of the uniform convergence , one can find a row from which the distance becomes very small for any at the same time. This condition is thus much stronger. In particular, a succession of functions which converges uniformly on a unit converges simply on this one. The reciprocal one is in general false except in very particular cases (see Théorèmes of Dini).
Uniform criterion of Cauchy
Now, one supposes in more than the metric Espace is a complete Espace. It is the case of good number of metric spaces, such as for example of real line provided with its absolute Value or more generally of all Space of Banach.
Under these conditions, one shows that a succession of functions converges uniformly on if and only if it checks the uniform criterion of Cauchy , namely:
As in the case of the series Cauchy, it is not necessary of exhiber the function towards which a succession of functions tends to show that convergence is uniform.
Uniform convergence of functions to values in a normalized vector space
It is supposed now that is a metric Espace and that is a vector Space normalized: it is a metric Espace from which topology is resulting from the distance such as:
There is the following fundamental result:
Proof. Is given. There exists an entirety such as, for all , . The function is continuous in any point . There exists thus open a container such as for all . Then, if ,
When is not compact, uniform convergence is a rare phenomenon. For example, converges uniformly towards on all compact of when the entirety tends towards the infinite one, but not on ; one whole Series of ray of convergence converges uniformly on very compact open disc of center 0 and ray , but one cannot say better in general.
In fact, continuity being a local property, uniform convergence on
" suffisamment" parts of is enough to ensure the continuity of the function limite.
- When is Localement compact, or when its topology is defined by metric.
- One with the same conclusion when is a space of Banach, if uniform convergence takes place
The following result, less extremely than the Theorem of dominated convergence, is also much less
difficult with montrer.
If is a Intervalle of , if or , then if a succession of functions integrable converges uniformly towards a function integrable then: .
Its use is at the base of the following result of Analyze complexes.
Is a succession of holomorphic functions on open of , converging uniformly on very compact of towards a function . Then is holomorphic.
The following notation is introduced:
It follows directly that a succession of functions converges uniformly towards a function if and only if:
: is in general not a standard on the vector Space of the functions of with values in .
Case where X is compact
One supposes from now on that X is a metric Espace compact, being always a vector Space normalized. One notes the whole of the continuous functions definite on and with values in .
Then: is a vector Space normalized. So moreover, are complete then is him also complete.
Space continuous numerical functions on
One chooses in this section a compact interval of and . Since provided with the absolute value is complete, it results from it that the vector Space normalized provided with the standard is complete.
Theorem of Weierstrass
The theorem of Weierstrass affirms that one can approach in a uniform way any function numerical continues on by a succession of very regular functions to knowing by polynomials. More precisely, if is a continuous function on then:
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