Fréchet Spaces

Introduction

A Fréchet space is a locally convex space which is metrisable and complete. The definition is short, and it isolates exactly the class of spaces in which the two most useful theorems of functional analysis hold simultaneously: the uniform boundedness principle, which needs the Baire category theorem, and the closed graph theorem, which needs the completeness. The function spaces that arise in analysis — the continuous functions on a compact space with the supremum norm, and the spaces of smooth, holomorphic and rapidly decreasing functions — are Fréchet spaces, and the spaces that are not are usually inductive limits of Fréchet spaces rather than genuinely different objects.

This article develops the theory of Fréchet spaces from the vocabulary fixed in Locally Convex Spaces. It treats the invariant metric and the $F$-norm, the characterisation of Fréchet spaces as the closed subspaces of countable products of Banach spaces, the Baire category theorem and its consequences, the open mapping and closed graph theorems, the boundedness and the Heine–Borel property of the Montel spaces, and the projective limit picture. It develops the standard examples of the class and names the analytic examples — the spaces of smooth, holomorphic and rapidly decreasing functions — whose definitions use the derivative and which are therefore Part III's, developed there rather than here. The analysis of the operators on these spaces, and the distribution theory that the examples support, belongs to Analysis on Linear Spaces in Part III; the general theory of locally convex spaces is that of Locally Convex Spaces, and the duality theory of the dual spaces that appear here is that of Duality Theory. Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, $E$ denotes a locally convex space over $\mathbb{K}$, and a Fréchet space is a complete metrisable locally convex space, in the notation of Locally Convex Spaces. A $F$-space is a complete metrisable topological vector space, not assumed locally convex. A sequence $(p_n)_{n \geq 0}$ of seminorms is increasing if $p_n \leq p_{n+1}$ pointwise.

Metrics on Locally Convex Spaces

The Invariant Metric

Definition. Let $(p_n)_{n \geq 0}$ be an increasing sequence of seminorms generating the topology of a metrisable locally convex space $E$. The $F$-norm associated with $(p_n)$ is

$$ \lvert x \rvert = \sum_{n \geq 0} 2^{-n} \min\bigl(1, p_n(x)\bigr) , $$

and the associated translation-invariant metric is $d(x,y) = \lvert x - y \rvert$.

Proposition. The $F$-norm has the following properties.

(a) $\lvert x \rvert \geq 0$ with $\lvert x \rvert = 0$ if and only if $p_n(x) = 0$ for all $n$; $\lvert \lambda x \rvert \leq \lvert x \rvert$ for $\lvert \lambda \rvert \leq 1$, and $\lvert \lambda x \rvert \to 0$ as $\lambda \to 0$.

(b) $\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert$, so $d$ is a metric, and it is translation-invariant: $d(x + z, y + z) = d(x,y)$.

(c) $d$ induces the topology generated by the seminorms $p_n$.

Proof. (a) is immediate from the definition and the properties of the seminorms. (b) follows from the subadditivity of each $p_n$ and the elementary inequality $\min(1, a+b) \leq \min(1,a) + \min(1,b)$, together with the homogeneity of the weights; translation invariance is the definition $d(x,y) = \lvert x - y\rvert$. (c) The balls of $d$ and the sets $U_{p_1, \dots, p_N; \varepsilon}$ define the same neighbourhood filter of $0$: a $d$-ball of radius $2^{-N-1}$ is contained in $\{p_1 < 1, \dots, p_N < 1\}$ because the weights outside the first $N$ terms sum to $2^{-N}$, and conversely $\{p_1 < \varepsilon_1, \dots, p_N < \varepsilon_N\} \subseteq \{d < 2^{-N} + \sum_{n \leq N}2^{-n}\varepsilon_n'\}$ for suitable small $\varepsilon_1, \dots, \varepsilon_N$.

Remark. The $F$-norm is not a norm unless the space is normable, and it is not homogeneous for general scalars: $\lvert \lambda x \rvert$ need not equal $\lvert \lambda \rvert \lvert x \rvert$. What survives is the continuity of scalar multiplication and the translation invariance, and these are enough for the completeness arguments. The choice of weights $2^{-n}$ is immaterial; any sequence of positive weights with finite sum gives the same topology.

Theorem (completeness of the metric). Let $E$ be a metrisable locally convex space with an increasing generating sequence $(p_n)$ and the metric $d$ above. Then $E$ is complete if and only if every sequence $(x_k)$ which is Cauchy for $d$ converges, equivalently if and only if $p_n(x_k - x_l) \to 0$ as $k, l \to \infty$ for every $n$ forces the existence of $x \in E$ with $p_n(x_k - x) \to 0$ for every $n$.

Proof. The metric $d$ and the family $(p_n)$ induce the same uniformity, so the two completeness conditions coincide; the statement is then the definition of completeness for the metric.

Fréchet Spaces as Projective Limits

Definition. Let $(E_n)$ be a sequence of Banach spaces and let $\pi_n : E_{n+1} \to E_n$ be continuous linear maps of norm at most $1$. The projective limit $\varprojlim_n E_n$ is the subspace of the product $\prod_n E_n$ consisting of the threads $(x_n)$ with $\pi_n(x_{n+1}) = x_n$ for all $n$, equipped with the subspace topology.

Theorem. A Hausdorff locally convex space $E$ is a Fréchet space if and only if it is isomorphic to a closed subspace of a countable product of Banach spaces; equivalently, if and only if it is isomorphic to a projective limit $\varprojlim_n E_n$ of a sequence of Banach spaces with linear transition maps.

Proof. If $(p_n)$ is an increasing generating sequence of seminorms, let $N_n = \{x : p_n(x) = 0\}$ and let $\widehat{E}_n$ be the completion of $E/N_n$ for the norm induced by $p_n$; the natural maps $E/N_{n+1} \to E/N_n$ extend to contractions of the completions, and the map $E \to \varprojlim_n \widehat{E}_n$ is a topological embedding whose image is closed when $E$ is complete, because a thread in the image is the limit of the images of a Cauchy sequence in $E$. Conversely a closed subspace of a product of countably many Banach spaces, with the product topology given by the countably many norm seminorms, is metrisable and complete, hence Fréchet, and being closed in a product of complete spaces it is complete.

Corollary. A Fréchet space is a closed subspace of a countable product of Banach spaces; consequently it inherits the completeness of the product, and its topology is determined by countably many seminorms, each of which may be taken to be a norm on a quotient.

Proof. This restates the theorem.

Example (the projective limit picture). The classical instances are analytic: the space $C^\infty([0,1])$ is the projective limit of the Banach spaces $C^n([0,1])$ with the $\mathcal{C}^n$-norm, and the Schwartz space is the projective limit of the weighted Sobolev spaces $H^{(m)}$. Both are Part III's, where the derivative is available, and the picture is recorded here as the standard illustration of the theorem.

The Baire Property and its Consequences

Baire Category

Theorem (Baire). Every complete metric space is a Baire space: a countable union of nowhere dense subsets has empty interior, equivalently a countable intersection of dense open subsets is dense.

Proof. Let $U_1, U_2, \dots$ be dense open subsets of a complete metric space $X$. Given a non-empty open $V_0$, choose a closed ball $B_1 \subseteq U_1 \cap V_0$ of radius less than $1$; having chosen $B_n$, choose a closed ball $B_{n+1} \subseteq U_{n+1} \cap B_n$ of radius less than $1/(n+1)$. The centres form a Cauchy sequence by the shrinking radii, and its limit lies in every $B_n$ by closedness, hence in $V_0 \cap \bigcap_n U_n$; thus the intersection is dense.

Corollary. Every Fréchet space is a Baire space, and hence is barrelled; consequently the Banach–Steinhaus theorem holds for families of continuous linear maps whose domain is a Fréchet space.

Proof. A Fréchet space is a complete metric space, so Baire's theorem applies; the passage from Baire to barrelledness is the standard argument: for a barrel $B$, the sets $E = \bigcup_n nB$ exhibit $E$ as a countable union of the closed sets $nB$, one of which has non-empty interior by Baire, and the convexity and balancedness of $B$ translate an interior point of $nB$ into a neighbourhood of $0$ contained in $B$.

The Open Mapping and Closed Graph Theorems

Theorem (open mapping). Let $T : E \to F$ be a continuous linear surjection of Fréchet spaces. Then $T$ is open: the image of every open set is open.

Proof. It suffices to show that $T$ maps a neighbourhood of $0$ in $E$ onto a neighbourhood of $0$ in $F$. Let $U$ be a balanced open neighbourhood of $0$ in $E$ with $U + U \subseteq 2U$. Since $T$ is surjective and $U$ is absorbing, $F = \bigcup_n T(nU) = \bigcup_n \overline{T(nU)}$; by the Baire property of $F$ some $\overline{T(nU)}$ has non-empty interior, and since it is convex and balanced it contains a balanced neighbourhood $W$ of $0$; scaling gives $W \subseteq \overline{T(U)}$. Now fix $y \in W$. Because $\overline{T(U/2^n)} \supseteq W/2^n$ for every $n$, one constructs recursively $x_n \in U/2^{n-1}$ with $y - T(x_1 + \cdots + x_n) \in W/2^n$: the step is possible because $y - T(x_1 + \cdots + x_{n-1})$ lies in $W/2^{n-1}$, whose half meets the image of $U/2^{n-1}$. The series $\sum_n x_n$ converges in the complete space $E$, its sum $x$ lies in $2U$, and $T x = y$ by continuity. Hence $T(2U) \supseteq W$, and $T$ is open.

Theorem (closed graph). Let $T : E \to F$ be a linear map between Fréchet spaces whose graph $\Gamma(T) = \{(x, Tx) : x \in E\}$ is closed in $E \times F$. Then $T$ is continuous.

Proof. Let $G = \Gamma(T)$, a closed subspace of the Fréchet space $E \times F$; it is therefore a Fréchet space. The two projections $\pi_1 : G \to E$ and $\pi_2 : G \to F$ are continuous and linear, and $\pi_1$ is a continuous bijection. By the open mapping theorem $\pi_1$ is a homeomorphism, so the composition $\pi_2 \circ \pi_1^{-1} = T$ is continuous.

Corollary. A linear map of Fréchet spaces is continuous if and only if it has closed graph; and a bijective continuous linear map of Fréchet spaces is a topological isomorphism.

Proof. The first is the closed graph theorem together with the evident continuity of the graph of a continuous map; the second is the open mapping theorem applied to the bijection and its inverse.

Boundedness and the Heine–Borel Property

Definition. A locally convex space $E$ is Montel if it is barrelled and every closed bounded subset is compact; equivalently, if it is barrelled and satisfies the Heine–Borel property that every closed bounded set is compact. A reflexive Montel space is a Montel space that is also reflexive; reflexivity, its relation to the Montel property and the strong dual are developed in Duality Theory.

Theorem. A Fréchet space is Montel if and only if every bounded sequence has a convergent subsequence; a Fréchet–Montel space is reflexive, and its strong dual is also a Montel space.

Pro. The equivalence uses the metrisability: a closed bounded set is compact exactly when every sequence in it has a convergent subsequence. Reflexivity of a Montel space is a theorem of the duality theory, and the dual statement is the closed-range theorem for Montel spaces; both are quoted as standard and are developed in Duality Theory.

Example (a non-Montel Fréchet space). A Banach space of infinite dimension is not Montel: its closed unit ball is bounded but is not compact, since a compact ball would give a finite-dimensional space. So the Heine–Borel property is a genuine restriction even among Fréchet spaces.

The Standard Examples

Spaces of Functions

Example ($C(X)$ for compact $X$). Let $X$ be a compact topological space. Then $C(X)$ with the supremum norm $\lVert f \rVert = \sup_X \lvert f \rvert$ is a Banach space, hence a Fréchet space; the topology is that of uniform convergence.

Example ($C(X)$ with compact convergence). Let $X$ be locally compact and $\sigma$-compact with an exhaustion $K_1 \subseteq K_2 \subseteq \cdots$ by compact sets. Then the seminorms $p_n(f) = \sup_{K_n}\lvert f \rvert$ are increasing and generate the topology of uniform convergence on compacta, so the space $C(X)$ is a Fréchet space; it is not normable when $X$ is not compact, by the Kolmogorov–von Neumann criterion of Locally Convex Spaces.

Example (the analytic function spaces, named). The space $C^\infty(U)$ of smooth functions on an open $U \subseteq \mathbb{R}^n$ with the seminorms of the derivatives on a compact exhaustion, the space $\mathcal{O}(\Omega)$ of holomorphic functions on a domain with the seminorms $\sup_{K_n}\lvert f \rvert$, and the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ of rapidly decreasing smooth functions with the seminorms $\sup_x\lvert x^\alpha \partial^\beta f(x)\rvert$ are Fréchet spaces, and $\mathcal{O}(\Omega)$ and $\mathcal{S}(\mathbb{R}^n)$ are Fréchet–Montel. Every one of them is defined by seminorms built from the derivatives of every order, so every one is an object of Part III, where the derivative and the integral are available; each is developed there, the Schwartz space and its dual, the tempered distributions, in Distributions and Fundamental Solutions and the operator theory on these spaces in Analysis on Linear Spaces. This Part records the structural point alone: the topologies of analysis are countable seminorm topologies, so the theory above applies to them, and the examples are named here only as instances of that. Two delimitations matter for the class. The space of compactly supported smooth functions with the inductive limit topology is not metrisable, hence not Fréchet; and a complete metrisable topological vector space need not be locally convex at all, the function spaces whose topology is convergence in a mean being the standard case, which is exactly the failure of the Hahn–Banach theorem and places them outside the theory.

Summary

A Fréchet space is a complete metrisable locally convex space; equivalently, a Hausdorff locally convex space whose topology is generated by a countable family of seminorms and which is complete for the induced uniformity. The topology may be given by the translation-invariant metric $d(x,y) = \sum_n 2^{-n}\min(1,p_n(x-y))$ for an increasing generating sequence $(p_n)$; the resulting $F$-norm is not homogeneous in general, but it is subadditive and compatible with scalar multiplication. A Fréchet space is isomorphic to a closed subspace of a countable product of Banach spaces, and equivalently to a projective limit of a sequence of Banach spaces; the general projective limit of Banach spaces is a Fréchet space, and every Fréchet space arises so.

By the Baire category theorem a Fréchet space is a Baire space, hence barrelled, so the Banach–Steinhaus theorem holds on it; the open mapping theorem says that a continuous linear surjection of Fréchet spaces is open; the closed graph theorem says that a linear map of Fréchet spaces with closed graph is continuous; and a continuous bijection of Fréchet spaces is a topological isomorphism. A Fréchet space is Montel when every closed bounded subset is compact, equivalently when every bounded sequence has a convergent subsequence; a Fréchet–Montel space is reflexive and has reflexive strong dual. The standard examples are the Banach spaces, the space $C(X)$ of continuous functions with the topology of compact convergence, the space $C^\infty(U)$ with the seminorms of all derivatives on a compact exhaustion, the space $\mathcal{O}(\Omega)$ of holomorphic functions, which is Fréchet–Montel by Montel's theorem, and the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ of rapidly decreasing smooth functions, which is Fréchet–Montel; the space $L^p[0,1]$ with $p<1$ is a complete metrisable topological vector space which is not locally convex, and $C_c^\infty(U)$ with its inductive limit topology is a complete locally convex space which is not metrisable.

The operator theory on these spaces, the Fourier transform and the theory of distributions, belong to Analysis on Linear Spaces in Part III; the duality theory of the dual spaces belongs to this Part.

Summary of Notation

Symbol	Meaning
$\mathbb{K}$	$\mathbb{R}$ or $\mathbb{C}$
$E$, $F$	Fréchet or locally convex spaces
$p_n$	An increasing generating sequence of seminorms
$\lvert x \rvert = \sum_n 2^{-n}\min(1,p_n(x))$	The $F$-norm
$d(x,y) = \lvert x - y \rvert$	Translation-invariant metric
$F$-space	Complete metrisable topological vector space
Fréchet space	Complete metrisable locally convex space
$\varprojlim_n E_n$	Projective limit of a sequence of Banach spaces
$C(X)$	Continuous functions, a Fréchet space for compact convergence
$\mathcal{S}(\mathbb{R}^n)$	Schwartz space, named here and developed in Part III
$s$	Space of rapidly decreasing sequences
$H_\beta$	Hermite polynomial of degree $\beta$
Montel	Barrelled with every closed bounded set compact
$\Gamma(T) \subseteq E \times F$	Graph of a linear map