Space of Fréchet

the space of Fréchet within the meaning of general topology is described with the article Espace T1. ---- A space of Fréchet is a mathematical structure of topological vector Space satisfying certain theorems relating to the spaces of Banach even in the absence of a standard. This denomination refers to Maurice Fréchet, mathematician French having taken part in particular in the foundation of the Topologie and with its applications in functional analyzes. It is in the latter field that the structure of spaces of Fréchet appear particularly useful, in particular by providing a natural topology to infinitely derivable spaces of functions.

Definition

A space of Fréchet is a topological vector Space real, complete within the meaning of uniform spaces and satisfying one of the two following equivalent conditions:

space is locally convex and métrisable by a invariant distance by translation;
there exists a countable family and separating from Semi-norme S continuous which generates the topology of the espaceballs” constitute a base of vicinities of the origin-->.

The equivalence of these two conditions is shown by building a family countable and separating from pseudo norms starting from any invariant distance and reciprocally. There is however no natural Bijection between the compatible invariant distances and the families countable and separating from pseudo norms.

For a space of Fréchet given, there exist several invariant distances in general defining topology and they induce a whole a structure of complete metric space. In the same way, there is no canonical choice of family of pseudo norms.

Examples

All Espace of Banach is a space of Fréchet but the reciprocal one is not always true. In particular, certain spaces of Fréchet are not normables.

It is the case of the C^∞ space () of the functions infinitely differentiable S on the interval, which can be provided with the pseudo norms for entire K ≥ 0:

_k = \ sup_ {} \ left (f^ {(K)}\ right)

F ||_K = sup

Properties

The assumption of complétude makes it possible to apply to spaces of Fréchet the Théorème of Baire and its consequences, inter alia:

the Theorem of Banach-Steinhaus: any simply limited family of applications of a space of Fréchet in a topological vector space is équicontinue;
the Theorem of the open application: any surjective continuous application between two spaces of Fréchet is open;
its corollary: any bijective continuous application between two spaces of Fréchet is a Homéomorphisme;
the Theorem of the graph closed: any application of graph closed between two spaces of Fréchet is continuous.

Local convexity ensures also the following properties:

the points of a space of Fréchet are separated by its topological Dual;
very convex compact of a space of Fréchet is the convex envelope of its extreme points.

The Théorème of local inversion does not apply in general to spaces of Fréchet, but a weak version was found under the name of Théorème of Nash-Moser.

Derived from Cakes

The space of the continuous linear applications between two spaces of Fréchet not constituting a priori a space of Fréchet, the construction of a differential for the continuous functions between two spaces of Fréchet passes by the definition of derived from Cakes.

That is to say Φ a function defined on open a U of a space of Fréchet X , with values in a space of Fréchet Y .
The derived from Cakes of Φ in a point X of U and in a direction H of X is the limit in Y (when it exists)
$\ Phi' (X; H) = \ lim_ {T \ to 0} \, \ frac {1} {T} \ Big (\ Phi (x+th) - \ Phi (X) \ Big)$ .
The function Φ is known as Cake-differentiable in X if there exists a continuous linear application Φ'_G ( X ) of X in Y such as for all H of X , (Φ'_G ( X ))( H ) = Φ' ( X; H ).

The differential of the Φ application can then be seen like a function defined on part of space of Fréchet X × X and with values in Y . It can possibly be differentiated in its turn.

For example, the linear operator of derivation D : C^∞ () → C^∞ () defined by D ( F ) = F “ is infinitely differentiable. Its first differential is for example defined for any couple ( F , H ) of infinitely derivable functions by Of ( F ) ( H ) = h' , in other words Of ( F ) = D .

However, the Théorème of Cauchy-Lipschitz does not extend to the resolution of the differential equations ordinary on spaces of Fréchet in any general information.

Equivalence of the two definitions

A succession of elements of a topological vector space is known as of Cauchy within the meaning of uniform spaces so for any vicinity of the origin, it exists a row from which the difference between two unspecified terms of the continuation is always in this vicinity. The vector space is known as complete if any continuation of Cauchy converges.

If there exists an invariant distance D whose balls constitute a base of open for a topological vector space locally convex E , this distance can be modified so that its balls is convex. The following applications of E in ℝ then train a family separating from continuous pseudo norms subscripted by the positive entireties:
$p_k \ colonist X \ mapsto \ inf \ left \ {\ lambda \ in \ R^+ \ colonist D \ left (0, \ frac {1} {\ lambda} X \ right) \ the \ frac {1} {k+1} \ right \}$ .
Any ball centered on the origin for the distance D thus contains the “ball unit” of the one of the pseudo norms.

If there exists a continuation separating from continuous pseudo norms ( p_k ) on a complete topological vector space E and who generates the topology of E , these standards can be modified so that the continuation is increasing. In this case, the balls of pseudo norms form a base of vicinities of the origin. The following application D then defines an invariant distance on E : $d \ colonist (X, there) \ mapsto \ sum_ {K} \ frac {2^ {- K} p_k (y-x)}{1+p_k (y-x)}$ .

If the assumption of local convexity is not satisfied, as on spaces L^p with p < 1, the existence of an invariant distance and supplements is not enough to define a structure of space to Fréchet.

See too

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