Simple convergence

The simple convergence or specific is a criterion of convergence in a functional space, i.e. in a whole of functions. It is a not very demanding criterion, consequently, in the event of convergence simple convergence is often checked. On the other hand, the passage in extreme cases offers much less properties than a stronger convergence like the uniform Convergence.

Definition

Simple convergence

That is to say (f_ {N}) _ {N} \, \! a succession of functions defined on X \, \! with values in Y \, \! . Lastly, is A \ subset X part of X \, \! . It is said that the continuation of functions (f_ {N}) _ {N} \, \! converges simply on A \, \! if:
\ forall X \ in A, the continuation (f_ {N} (X))_ {N} \, \! converges in Y \, \!

  • If one notes f (X) = \ lim_ {N \ rightarrow + \ infty} f_ {N} (X) one says whereas the continuation of functions (f_ {N}) _ {N} \, \! converges simply on A \, towards the function f \, .

Notice

In this definition, one supposed the topological Espace Y \, separate. One can justify such a choice by the fact that in a separated space, if a succession of elements of this space converges then necessarily its limit is single (what is not the case in a topological space not-separate).

The unicity of the limit is thus an essential condition to be able to define the simple convergence of a succession of functions towards a function.

Weak topology

Definition

There exists a Topologie associated with simple convergence, one calls in general it weak topology. This topology is often defined using a bases vicinities. It in the following way is defined:

That is to say f \; a function of X \, \! in Y \, \! two topological spaces such as Y \, \! is separate. That is to say x \, \! an element of X \, \! such as f \; is defined in x \, \! . One considers then \ mathcal V (F (X)) a base of vicinity of f (X) \, \! for the topology of Y \, \! . With each element V_ {F (X)}\; of \ mathcal V (F (X)) one associates the subset W_ {(F, X)}\; of the functions \ phi \; of X \, \! in Y \, \! defined in x \, \! and such as \ phi (X) \; is element of V_ {F (X)}\; . Union of all the whole of the type W_ {(F, X)}\; when f \; traverses the whole of the functions and x \, \! traverses the field of definition of f \; forms a base of vicinity. Associated topology is called weak topology.

Remarks

It is relatively simple to show that the simple convergence of a succession of functions (f_n) _n \; is equivalent to convergence for the weak topology of the continuation.

If X \, \! is not a finished unit, then it does not exist distance associated with this topology. We know indeed that any metric space is provided with a deduced topology. This topology can never be weak topology.

Properties

Weak topology is a not very constraining criterion of convergence as its name indicates it. There thus exist less properties than in the case of uniform convergence for example.

  • uniform convergence implies simple convergence. The demonstration rises directly from the definitions. On the other hand the reciprocal one is false as the counterexample illustrated graphically at the beginning of article shows it.

  • simple convergence does not preserve continuity, as the counterexample illustrated graphically at the beginning of article shows it.

  • If the starting whole is a measurable Espace and where the whole of arrival is the body of the real then simple convergence can indicate convergence for the standard L_1 with the addition of certain assumptions described in the articles monotonous Théorème of convergence and Théorème of dominated convergence.

  • the preceding result is true only within the framework of the Intégrale of Lebesgue and not in that of Riemann.

Simple convergence in a metric space

It is supposed now that Y \, is a metric Espace, i.e. Y \, is provided with a distance d \, and topology which is associated for him. It is known initially that a metric space is always separate. One can then translate the concept of convergence simple into terms of “epsilon”:

A succession of functions (f_ {N}) _ {N} \, converges simply on A \, towards a function f \, if and only if:

\ forall X \ in has, \ forall \ epsilon >0, \ exists N_ {\ epsilon, X}, \ forall N \ in \ NR, N \ Ge N_ {\ epsilon, X} \ Rightarrow D (f_ {N} (X), F (X))< \ epsilon

See too

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