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
$$d:U\times U\to \mathbb{R}$$ |
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 \mathbb{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.
References
- 1 W.Rudin, Functional Analysis^{}.
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 |