Fréchet space - Imported Knowlpedia Content

Definition

A Fréchet space is a Hausdorff locally convex topological vector space whose topology is induced by a translation-invariant metric with respect to which the space is complete. Equivalently, it is a complete, metrizable, locally convex space. Completeness belongs to the compatible uniform structure and is independent of which compatible translation-invariant metric is chosen. A Fréchet space need not be normable: one countable family of seminorms may be required to describe its topology, and no single norm may generate the same open sets.

Seminorm description

The topology of a Fréchet space can be generated by a countable separating family of seminorms $(p_n)_{n\geq1}$ . One compatible translation-invariant metric is

d(x,y)=\sum_{n=1}^{\infty}2^{-n} \frac{p_n(x-y)}{1+p_n(x-y)}.

Conversely, a Hausdorff locally convex space is metrizable exactly when its topology is generated by a countable family of seminorms; it is Fréchet when it is complete for the resulting uniform structure.

Examples

Every Banach space is Fréchet. The space $C^\infty(M)$ on a compact smooth manifold is Fréchet for seminorms controlling derivatives of increasing order. The Schwartz space $\mathcal S(\mathbb R^n)$ is another basic example. These spaces are usually not normable because convergence must simultaneously control infinitely many derivatives or weighted derivatives.

Mapping theorems and terminology

The Baire property of complete metric spaces gives Fréchet spaces versions of the open mapping, closed graph, and uniform boundedness theorems. These results make the category much more rigid than arbitrary locally convex spaces.

Terminology warning. Some authors use F-space for any complete metrizable topological vector space, without local convexity. Under that convention, every Fréchet space is an F-space but not conversely.

References

Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: Chapter 1.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Elsevier book record. Relevant: Chapter 9.