Fréchet space

We consider two classes of topological vector spaces, 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 topology is induced by a translation invariant metric. To be more precise, we say that $U$ is an F-space if there exists a metric function

$$d:U\times U\to ℝ$$

such that

$$d(x,y)=d(x+z,y+z),x,y,z\in U;$$

and such that the collection of balls

is a base for the topology of $U$.

Note 1.

Recall that a topological vector space is a uniform space. The hypothesis that $U$ is complete is formulated in reference to this uniform structure. To be more precise, we say that a sequence $a_n\in U,n=1,2,\mathrm{\dots }$ is Cauchy if for every neighborhood $O$ of the origin there exists an $N\in \mathbb{N}$ such that $a_n-a_m\in O$ for all $n,m>N$. The completeness condition then takes the usual form of the hypothesis that all Cauchy sequences possess a limit point.

Note 2.

It is customary to include the hypothesis that $U$ is Hausdorff 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 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.

Definition 2.

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

$$\parallel -{\parallel }_n:U\to ℝ,n\in \mathbb{N},$$

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 converse. 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

$$d(x,y)=\sum _{n=0}^{\mathrm{\infty }}{2}^{-n}\frac{{\parallel x-y\parallel }_n}{1+{\parallel x-y\parallel }_n},x,y\in U.$$ (1)

We now show that $d$ satisfies the metric axioms. Let $x,y\in U$ such that $x\ne y$ be given. Since $U$ is Hausdorff, there is at least one seminorm such

$${\parallel x-y\parallel }_n>0.$$

Hence $d(x,y)>0$.

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

$$a\le b+c.$$

A straightforward calculation shows that

$$\frac{a}{1+a}\le \frac{b}{1+b}+\frac{c}{1+c},$$ (2)

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

$${\parallel x-z\parallel }_n\le {\parallel x-y\parallel }_n+{\parallel y-z\parallel }_n,x,y,z\in U$$

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

Next let us suppose that $U$ is a locally convex F-space, and prove that it is Fréchet. For every $n=1,2,\mathrm{\dots }$ let $U_n$ be an open convex neighborhood of the origin, contained inside a ball of radius $1/n$ about the origin. Let $\parallel -{\parallel }_n$ be the seminorm with $U_n$ 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