Involutive Fréchet Spaces

Introduction

A Fréchet space is complete and metrisable, and on it the notion of an involution is governed by two facts that fail in a general locally convex space. A Fréchet space is a Baire space, so a linear involution with closed graph is continuous; and it is bornological, so a linear involution that carries bounded sets to bounded sets is continuous, and an involution is continuous exactly when it is bounded. The topology may be generated by an increasing sequence of invariant seminorms, the fixed and negated subspaces are closed and are Fréchet spaces for the induced structure, the splitting into them is topological, and on the strong dual — which is complete but in general not metrisable — the transposed involution is continuous with the fixed and negated parts given by the annihilator calculus.

This article develops the theory of an involution on a Fréchet space. The locally convex background and the invariant seminorms are Locally Convex Spaces and Locally Convex Spaces with an Involution; the metrisability, the completeness, the $F$-norm and the closed graph theorem for Fréchet spaces are Fréchet Spaces; the bounded sets, the bornological property and the closed graph theorem are Bounded Sets and Bornological Spaces and The Closed Graph Theorem; the strong dual and the annihilators are Duality Theory. The Banach case is Involutive Banach Spaces, the nuclear refinement is Involutive Nuclear Spaces, and the completed tensor products are The Involution on a Topological Tensor Product. No form and no Hilbert structure is used.

Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, $\varsigma$ is the involution of $\mathbb{K}$, $E$ is a Fréchet space over $\mathbb{K}$ with an increasing generating sequence of seminorms $(p_{n})$, and $\theta$ is a $\varsigma$-semilinear involution of $E$ with $\theta^{2} = \mathrm{id}$; the linear case is written $T$, the fixed and negated subspaces are $E^{\theta}$ and $E^{-}$, $\lvert x\rvert = \sum_n 2^{-n}\min(1, p_n(x))$ is the $F$-norm, and $E'_{b}$ is the strong dual.

Continuity and Boundedness

Theorem (the two criteria). For a $\varsigma$-semilinear involution $\theta$ of the Fréchet space $E$ the following are equivalent:

$\theta$ is continuous;
$\theta$ has closed graph;
$\theta$ is bounded, that is, it carries bounded sets to bounded sets;
for every $n$ there are $m$ and a constant $C$ with $p_{n}(\theta x) \leq C\,p_{m}(x)$ for all $x$.

Proof. $(1) \Rightarrow (2)$ a continuous map has closed graph; $(2) \Rightarrow (1)$ the closed graph theorem for Fréchet spaces, which are $F$-spaces, of The Closed Graph Theorem; $(1) \Rightarrow (3)$ a continuous linear map carries bounded sets to bounded sets; $(3) \Rightarrow (1)$ the bornological property of a metrisable locally convex space, from Bounded Sets and Bornological Spaces, since a bounded linear map from a bornological space is continuous; and $(1) \Leftrightarrow (4)$ is the seminorm criterion of Locally Convex Spaces with an Involution.

Proposition (the invariant $F$-norm). For a continuous involution the formula $\lvert x\rvert_{\theta} = \lvert x\rvert + \lvert\theta x\rvert$ is an $F$-norm equivalent to $\lvert\cdot\rvert$, it is $\theta$-invariant, $\lvert\theta x\rvert_{\theta} = \lvert x\rvert_{\theta}$, and it satisfies $\lvert\theta x\rvert \leq \lvert x\rvert_{\theta}$; the same holds for the $F$-norm built from the invariant seminorms $q_{n}$.

Proof. The sum of two $F$-norms is an $F$-norm equivalent to each, and invariance is $\lvert\theta x\rvert + \lvert\theta^{2}x\rvert = \lvert\theta x\rvert + \lvert x\rvert$; the estimate is the definition. The statement for the $q_{n}$ is the analogous computation with the invariant seminorms of Locally Convex Spaces with an Involution.

Invariant Seminorms and the F-Norm

Theorem (invariant generating sequence). Let $\theta$ be a continuous involution of $E$. Then the seminorms

$$ q_{n} = \sup(p_{n}, p_{n}\circ\theta) $$

form an increasing generating sequence of $\theta$-invariant seminorms defining the topology, so the Fréchet structure may be given by invariant seminorms; with respect to them $\theta$ is isometric, and the metric $d_{\theta}(x,y) = \sum_n 2^{-n}\min(1, q_n(x-y))$ is $\theta$-invariant.

Proof. The suprema generate the topology and are invariant by Locally Convex Spaces with an Involution; they are increasing with $n$ because the $p_{n}$ are, so they form a generating sequence of Fréchet Spaces, and the invariance $q_{n}\circ\theta = q_{n}$ makes the metric invariant.

Proposition (bounded sets are the same). The bounded sets of $E$ are the bounded sets of the invariant family $(q_{n})$, and a $\theta$-stable bounded set is carried to a bounded set; on a Banach space this reduces to the statement that a continuous involution is bounded with norm $\lVert\theta\rVert$.

Proof. The equivalence of the seminorm families preserves the bounded sets, by Fréchet Spaces; the invariance shows that $\theta$ preserves a $\theta$-stable bounded set.

The Fixed and Negated Subspaces

Theorem (the summands are Fréchet). The fixed subspace $E^{\theta}$ and the negated subspace $E^{-}$ are closed subspaces of $E$, hence Fréchet spaces for the induced structure and the restricted generating seminorms; in the antilinear case $E^{\theta}$ is a real Fréchet space. When $2$ is invertible the averaging maps $\pi_{\pm} = \frac12(\mathrm{id}\pm\theta)$ are continuous projections and

$$ E = E^{\theta} \oplus E^{-} $$

is a topological direct sum of Fréchet spaces.

Proof. Closedness is Involutive Topological Linear Spaces; a closed subspace of a complete metrisable locally convex space is complete and metrisable, hence Fréchet, by Fréchet Spaces; the projections and the splitting are Locally Convex Spaces with an Involution, and the open mapping theorem for Fréchet spaces gives the equivalence of the product structure with the given one.

Proposition (the fixed subspace and the closure). The completion of the fixed subspace is the fixed subspace, $E^{\theta}$ being closed; for a dense involutive subspace $D \subseteq E$ the involution of $E$ is the extension of the involution of $D$ and $E^{\theta} = \overline{D^{\theta}}$.

Proof. The direction $\supseteq$ is continuity and closedness, and $\subseteq$ is the density argument of Involutive Banach Spaces, carried over verbatim to Fréchet spaces using the boundedness of $\pi_{+}$.

The Strong Dual

Theorem (the transposed involution on the strong dual). The strong dual $E'_{b}$ is a complete locally convex space, the transposed involution $\theta'(\varphi) = \varphi\circ\theta$ is a continuous involution of the same kind, and its fixed and negated subspaces are the annihilators

$$ (E'_{b})^{\theta'} = (E^{-})^{\circ}, \qquad (E'_{b})^{-} = (E^{\theta})^{\circ} , $$

which are weakly-star closed subspaces of $E'_{b}$; the transposed involution is an isometry of the strong topology onto itself.

Proof. The transposed involution is continuous for the strong topology by Locally Convex Spaces with an Involution; the annihilator identification is The Involution and the Dual Pairing, and the strong dual is complete by Duality Theory; the homeomorphism of the strong topology is the strong continuity of the transpose of The Dual Operator and the Weak Topology.

Proposition (the strong dual is polar-reflexive). When $E$ is reflexive the transposed involution of $\theta'$ on $E'' = E$ is $\theta$; the dual pair $(E, E'_{b})$ with the two involutions is a dual pair of involutive spaces, and the operation $\theta \mapsto \theta'$ is an involution of the reflexive involutive Fréchet spaces.

Proof. The symmetry $(\theta')' = \theta$ is The Involution and the Dual Pairing, and the reflexivity is Duality Theory.

Examples

Example (the Schwartz space). On the Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ of Fréchet Spaces, the map $f \mapsto \overline{f}$ is an antilinear isometric involution for each of the seminorms $\sup_{|\alpha|,|\beta|\le n}\lVert x^{\alpha}\partial^{\beta}f\rVert$, with fixed subspace the real Schwartz space $\mathcal{S}(\mathbb{R}^{n},\mathbb{R})$; the reflexion $f(x) \mapsto f(-x)$ is a linear isometric involution with fixed subspace the even functions, and the two commute.

Example (the space of continuous functions). On $C(X)$ with the topology of compact convergence the map $f \mapsto \overline{f}$ is an antilinear isometric involution with fixed subspace the real-valued continuous functions, a real Fréchet space; the map $f \mapsto f \circ \sigma$ for an involutive homeomorphism $\sigma$ of $X$ is a linear isometric involution with the $\sigma$-invariant functions as fixed subspace.

Example (the rapidly decreasing sequences). On the space $s$ of rapidly decreasing sequences the coordinatewise conjugation is an antilinear isometric involution with fixed subspace the real rapidly decreasing sequences, and the coordinatewise negation on the even coordinates is a linear isometric involution whose fixed subspace is the space of sequences vanishing on those coordinates.

Summary

On a Fréchet space a $\varsigma$-semilinear involution is continuous exactly when it is bounded, when it has closed graph, and when it satisfies the seminorm estimate $p_{n}(\theta x) \leq C p_{m}(x)$; this is the conjunction of the closed graph theorem for $F$-spaces and the bornological property of metrisable locally convex spaces, and it is strictly stronger than the locally convex criterion because the graph and the boundedness give continuity for free. The topology may be generated by an increasing sequence of invariant seminorms $q_{n} = \sup(p_{n}, p_{n}\circ\theta)$, with an invariant metric and an equivalent invariant $F$-norm, and the bounded sets are unchanged. The fixed and negated subspaces are closed, hence Fréchet for the induced structure, and when $2$ is invertible the space is the topological direct sum $E^{\theta}\oplus E^{-}$; a dense involutive subspace has the involution of the whole space as its extension and the fixed subspace of the whole space as the closure of its fixed subspace. On the strong dual, which is complete, the transposed involution is continuous with the fixed and negated parts the annihilators of the summands, and the operation is an involution of the reflexive involutive Fréchet spaces. The Schwartz space, the space of continuous functions with compact convergence and the space of rapidly decreasing sequences are the standard examples. The nuclear refinement is Involutive Nuclear Spaces.

Summary of Notation

Symbol	Meaning
$E$, $(p_{n})$	Fréchet space and an increasing generating sequence of seminorms
$\theta$, $T$	$\varsigma$-semilinear involution; the linear case
$\lvert x\rvert_{\theta}$	Equivalent invariant $F$-norm
$q_{n} = \sup(p_{n}, p_{n}\circ\theta)$	Invariant generating seminorms
$E^{\theta}$, $E^{-}$	Closed summands, Fréchet spaces
$\pi_{\pm} = \frac12(\mathrm{id}\pm\theta)$	Continuous averaging projections
$E'_{b}$	Strong dual, complete
$\theta'(\varphi) = \varphi\circ\theta$	Transposed involution on the strong dual
$(E^{-})^{\circ}$, $(E^{\theta})^{\circ}$	Annihilators, the dual summands