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

 $d:U\times U\rightarrow\mathbb{R}$

such that

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

and such that the collection of balls

 $B_{\epsilon}(x)=\{y\in U:d(x,y)<\epsilon\},\quad x\in U,\;\epsilon>0$

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 $a_{n}\in U,\;n=1,2,\ldots$ 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

 $\|-\|_{n}:U\rightarrow\mathbb{R},\quad n\in\mathbb{N},$

such that the collection of all balls

 $B_{\epsilon}^{(n)}(x)=\{y\in U:\|x-y\|_{n}<\epsilon\},\quad x\in U,\;\epsilon>% 0,\;n\in\mathbb{N},$

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

 $d(x,y)=\sum_{n=0}^{\infty}2^{-n}\frac{\|x-y\|_{n}}{1+\|x-y\|_{n}},\quad x,y\in U.$ (1)

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

 $\|x-y\|_{n}>0.$

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

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

 $a\leq b+c.$

A straightforward calculation shows that

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

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

 $\|x-z\|_{n}\leq\|x-y\|_{n}+\|y-z\|_{n},\quad x,y,z\in U$

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 $n=1,2,\ldots$ let $U_{n}$ 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 $U_{n}$ as the unit ball. By definition, the unit balls of these seminorms give a neighborhood base for the topology of $U$. QED.

References

Title Fréchet space FrechetSpace 2013-03-22 13:06:10 2013-03-22 13:06:10 rmilson (146) rmilson (146) 51 rmilson (146) Definition msc 57N17 msc 54E50 msc 52A07 TopologicalVectorSpace HausdorffSpaceNotCompletelyHausdorff F-space