Fréchet space

We consider two classes of topological vector spacesMathworldPlanetmath, one more general than the other. Following Rudin [1] we will define a Fréchet space to be an element of the smaller class, and refer to an instance of the more general class as an F-space. After giving the definitions, we will explain why one definition is stronger than the other.

Definition 1.

An F-space is a complete topological vector space whose topologyMathworldPlanetmathPlanetmath is induced by a translationMathworldPlanetmathPlanetmath invariant metric. To be more precise, we say that U is an F-space if there exists a metric function


such that


and such that the collectionMathworldPlanetmath of balls


is a base for the topology of U.

Note 1.

Recall that a topological vector space is a uniform space. The hypothesisMathworldPlanetmathPlanetmath that U is completePlanetmathPlanetmathPlanetmathPlanetmath is formulated in reference to this uniform structure. To be more precise, we say that a sequence anU,n=1,2, is Cauchy if for every neighborhood O of the origin there exists an N such that an-amO for all n,m>N. The completeness condition then takes the usual form of the hypothesis that all Cauchy sequencesPlanetmathPlanetmath possess a limit pointMathworldPlanetmathPlanetmath.

Note 2.

It is customary to include the hypothesis that U is HausdorffPlanetmathPlanetmath in the definition of a topological vector space. Consequently, a Cauchy sequence in a complete topological space will have a unique limit.

Note 3.

Since U is assumed to be complete, the pair (U,d) is a complete metric space. Thus, an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath definition of an F-space is that of a vector spaceMathworldPlanetmath equipped with a complete, translation-invariant (but not necessarily homogeneousPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( metric, such that the operationsMathworldPlanetmath of scalar multiplication and vector addition are continuousMathworldPlanetmathPlanetmath with respect to this metric.

Definition 2.

A Fréchet space is a complete topological vector space (either real or complex) whose topology is induced by a countableMathworldPlanetmath family of semi-norms. To be more precise, there exist semi-norm functions


such that the collection of all balls


is a base for the topology of U.

Proposition 1

Let U be a complete topological vector space. Then, U is a Fréchet space if and only if it is a locally convex F-space.

Proof. First, let us show that a Fréchet space is a locally convex F-space, and then prove the converseMathworldPlanetmath. Suppose then that U is Fréchet. The semi-norm balls are convex; this follows directly from the semi-norm axioms. Therefore U is locally convex. To obtain the desired distance function we set

d(x,y)=n=02-nx-yn1+x-yn,x,yU. (1)

We now show that d satisfies the metric axioms. Let x,yU such that xy be given. Since U is Hausdorff, there is at least one seminorm such


Hence d(x,y)>0.

Let a,b,c>0 be three real numbers such that


A straightforward calculation shows that

a1+ab1+b+c1+c, (2)

as well. The above trick underlies the definition (1) of our metric function. By the seminorm axioms we have that


for all n. Combining this with (1) and (2) yields the triangle inequalityMathworldMathworldPlanetmath for d.

Next let us suppose that U is a locally convex F-space, and prove that it is Fréchet. For every n=1,2, let Un be an open convex neighborhood of the origin, contained inside a ball of radius 1/n about the origin. Let -n be the seminorm with Un as the unit ball. By definition, the unit balls of these seminorms give a neighborhood base for the topology of U. QED.


Title Fréchet space
Canonical name FrechetSpace
Date of creation 2013-03-22 13:06:10
Last modified on 2013-03-22 13:06:10
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 51
Author rmilson (146)
Entry type Definition
Classification msc 57N17
Classification msc 54E50
Classification msc 52A07
Related topic TopologicalVectorSpace
Related topic HausdorffSpaceNotCompletelyHausdorff
Defines F-space