We consider two classes of topological vector spaces, one more general than the other. Following Rudin  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.
Recall that a topological vector space is a uniform space. The hypothesis that is complete is formulated in reference to this uniform structure. To be more precise, we say that a sequence is Cauchy if for every neighborhood of the origin there exists an such that for all . The completeness condition then takes the usual form of the hypothesis that all Cauchy sequences possess a limit point.
It is customary to include the hypothesis that is Hausdorff in the definition of a topological vector space. Consequently, a Cauchy sequence in a complete topological space will have a unique limit.
Since is assumed to be complete, the pair is a complete metric space. Thus, an equivalent definition of an F-space is that of a vector space equipped with a complete, translation-invariant (but not necessarily homogeneous (http://planetmath.org/NormedVectorSpace)) metric, such that the operations of scalar multiplication and vector addition are continuous with respect to this metric.
A Fréchet space is a complete topological vector space (either real or complex) whose topology is induced by a countable 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 .
Let be a complete topological vector space. Then, 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 converse. Suppose then that is Fréchet. The semi-norm balls are convex; this follows directly from the semi-norm axioms. Therefore is locally convex. To obtain the desired distance function we set
Let be three real numbers such that
A straightforward calculation shows that
as well. The above trick underlies the definition (1) of our metric function. By the seminorm axioms we have that
Next let us suppose that is a locally convex F-space, and prove that it is Fréchet. For every let be an open convex neighborhood of the origin, contained inside a ball of radius about the origin. Let be the seminorm with as the unit ball. By definition, the unit balls of these seminorms give a neighborhood base for the topology of . QED.
- 1 W.Rudin, Functional Analysis.
|Date of creation||2013-03-22 13:06:10|
|Last modified on||2013-03-22 13:06:10|
|Last modified by||rmilson (146)|