Involutive Topological Linear Spaces

Introduction

An involutive topological linear space is a topological linear space together with a continuous involution of it. A topological linear space is a topological module over a topological ring $R$, commutative with $1 \neq 0$; over a topological field it is a topological vector space. The involution of the space is measured against an involution of the scalars: $\varsigma$ is a continuous involution of $R$, so that $R$ is an involutive topological ring in the sense of Involutive Topological Rings and Fields, and a map $\theta : V \to V$ is $\varsigma$-semilinear when

$$ \theta(\lambda x + \mu y) = \varsigma(\lambda)\theta(x) + \varsigma(\mu)\theta(y) \qquad \text{for all } \lambda, \mu \in R, \ x, y \in V, $$

and a $\varsigma$-semilinear involution when moreover $\theta^2 = \mathrm{id}$. When $\varsigma = \mathrm{id}$ the map is $R$-linear and the pair is a linear involution; when $\varsigma \neq \mathrm{id}$ the map is not $R$-linear, and over $\mathbb{C}$ with the complex conjugation, the case the article develops, it is called antilinear. The linear case is the topological form of the order-two operators of Involutive Linear Spaces, the semilinear case is the topological form of its semilinear involutions, and over $\mathbb{C}$ the antilinear case with $\theta^2 = \mathrm{id}$ is a real structure.

Four additions are the substance of the article. The fixed and the negated subspaces are closed, being equalizers of continuous maps into a Hausdorff space; the fixed subspace of a semilinear involution is moreover a closed topological module over the fixed subring $R^\varsigma$ of the scalars. When $2$ is invertible the averaging maps are continuous and $V$ is the topological direct sum $V^\theta \oplus V^-$, so the decomposition of the abstract theory survives as a topological splitting. For a linear topology the involution is continuous exactly when it almost preserves the filtration, and for the $I$-adic topology a linear involution is continuous automatically, so continuity restricts only the semilinear ones. And a continuous involution extends to the completion, where its fixed subspace is the closure of the fixed subspace of $V$.

Continuity is a hypothesis, and on a linear space it is free more often than on a ring. It is free for the identity; it is free on a finite-dimensional Hausdorff space over a complete valued field, where every linear map is continuous by Topological Modules and Vector Spaces; and it is free for the $I$-adic topology when the involution is linear. It fails in general, and the failure is detectable: on $\mathbb{C}[x]$ with the $(x)$-adic topology the involutions $f(x) \mapsto f(b - x)$ and $f(x) \mapsto \overline{f(b - x)}$ are continuous exactly for $b = 0$, and their fixed subspaces for $b \neq 0$ are dense and not closed. A linear involution can even be discontinuous for a linear topology that is not $I$-adic.

Layout and boundaries. The article has six sections and a closing comparison: the continuous involutions and their two kinds; the criterion for a linear topology; the fixed and the negated subspaces with the topological splitting; quotients and products; the completion; and the semilinear case with the fixed field. The abstract theory is Involutive Linear Spaces, whose fixed and negated summands, type, semilinear involutions and two signs of the antilinear square are used without repetition; the topological background is Topological Modules and Vector Spaces, whose neighbourhoods of zero, linear topologies, quotient topology, completion $\widehat{V} = \varprojlim_n V/U_n$ and Hausdorffness criterion are used with its meanings, together with the uniform completion of Topological Groups and the continuous ring involution, the closed fixed subring and the closed fixed field of Involutive Topological Rings and Fields. Three boundaries are marked rather than crossed. The norms, the operator norm and the equivalence of norms are Normed and Banach Spaces, the next article in this category, and no norm is written below. The weak and the strong topologies on the dual are Duality Theory, and the completed tensor products are Topological Tensor Products, both later in this category: the dual appears only as a linear space, with no topology on it, and no tensor product is topologised. The forms, the adjoints and the involutions a form induces are Hilbert Algebras, and the absolute values, the valuations and the valued fields are Absolute Values, Valuations and Completions, used only through the finite-dimensional proposition cited above. Throughout, $R$ is a topological ring, commutative with $1 \neq 0$, $\varsigma$ is a continuous involution of $R$, $V$ is a topological $R$-module, Hausdorff when a closedness claim is made, $\theta$ is a $\varsigma$-semilinear involution of $V$, $T$ is the name of a linear one, $R^\varsigma$ is the fixed subring, $V^\theta = \ker(\theta - \mathrm{id})$ and $V^- = \ker(\theta + \mathrm{id})$, $q$ is a quotient map, $\iota : V \to \widehat{V}$ is the map into the completion, and $2$ is invertible in $R$ whenever the averaging map is used.

Continuous Involutions

Definition and First Properties

Definition. A topological involution of the topological $R$-module $V$, relative to the involution $\varsigma$ of $R$, is a $\varsigma$-semilinear involution of $V$ that is continuous for the topology. An involutive topological linear space is a pair $(V,\theta)$ consisting of a topological linear space and a topological involution of it; over a topological field it is an involutive topological vector space. A morphism $(V,\theta) \to (W,\omega)$ is a continuous $R$-linear map $f : V \to W$ with $f \circ \theta = \omega \circ f$. When $\varsigma = \mathrm{id}$ the involution is linear and is written $T$; when $\varsigma \neq \mathrm{id}$ it is semilinear, and antilinear in the case of the complex conjugation.

Proposition (continuity at the origin). Let $\theta$ be a $\varsigma$-semilinear involution of $V$. Then $\theta$ is continuous if and only if it is continuous at $0$; a topological involution is a homeomorphism with $\theta^{-1} = \theta$, and it is uniformly continuous for the additive uniformity of $V$.

Proof. $\theta$ is additive with $\theta(0) = 0$, so continuity at every point is continuity at $0$; $\theta^2 = \mathrm{id}$ exhibits $\theta$ as a bijection whose inverse is itself, so a continuous involution is a homeomorphism. An additive continuous map is a continuous homomorphism of the additive topological group, and the image of a neighbourhood $U$ of $0$ is a neighbourhood of $0$, so $\theta$ carries the base set $E_U$ of the left uniformity into $E_{\theta(U)}$: it is uniformly continuous for the left uniformity, and likewise for the right one, the two being interchanged by the negation.

Remark (automatic continuity). Two hypotheses replace continuity. On a compact Hausdorff module an additive involution with closed graph is continuous, because the graph is then a closed subspace of the compact product and projects homeomorphically onto the first factor. And a Baire-measurable homomorphism of Polish groups is continuous, by Pettis's theorem, named and not used. Neither hypothesis is removable: on the $\mathbb{Q}$-vector space $\mathbb{R}$ with its usual topology a $\mathbb{Q}$-linear involution that exchanges two elements of a Hamel basis is discontinuous.

Remark (the two kinds, and what the topology sees). A linear involution reverses nothing: the additive group of $V$ is abelian, so it is at once an automorphism and an anti-automorphism of that group, and its fixed and negated subspaces are $R$-submodules. A semilinear involution does not act on the scalars $R$ but on the fixed subring $R^\varsigma$: if $x$ is fixed and $r \in R^\varsigma$ then $\theta(rx) = \varsigma(r)\theta(x) = rx$, so the fixed and the negated subspaces are $R^\varsigma$-submodules and not $R$-submodules in general. The topology does not remove the distinction, and the closedness theorem below gives it a topological form, a topological module structure over the closed subring $R^\varsigma$.

Remark (the transpose, named and not used). A continuous involution preserves the continuous dual: if $\varphi : V \to R$ is continuous and $R$-linear then so is $\varphi \circ \theta$, and $\theta'(\varphi) = \varphi \circ \theta$ is an involution of the continuous dual $V'$, of the same kind as $\theta$. This is a purely linear statement, the dual carrying no topology in this article; the weak and the strong topologies on it are Duality Theory, later in this category, and the annihilator calculus that goes with them is not used.

Examples

Example (the identity, verdict: a linear, continuous involution; the trivial case). The identity of $V$ is a topological involution, linear, with $V^{\mathrm{id}} = V$ and $V^- = 0$. It is the involution of every topological linear space that carries only the structure of its scalars.

Example (finite dimension over $\mathbb{R}$ and $\mathbb{C}$, verdict: every linear and every antilinear involution is continuous). Let $V$ be finite-dimensional and Hausdorff over a complete valued field. Then every linear map $V \to V$ is continuous, by the proposition of Topological Modules and Vector Spaces on finite-dimensional spaces, so every linear involution of $V$ is a topological involution. Over $\mathbb{C}$ the conjugation fixes $\mathbb{R}$ pointwise, so an antilinear map is $\mathbb{R}$-linear; the underlying real space is finite-dimensional over the complete valued field $\mathbb{R}$, so every antilinear involution is continuous too. In finite dimension over these fields the topological category therefore adds nothing to the abstract classification of Involutive Linear Spaces, and the linear and the antilinear kinds are both present and both continuous. The interesting topological phenomena need infinite dimension.

Example (the conjugation of $\mathbb{C}^n$, verdict: an antilinear, continuous involution whose fixed subspace is the real form). On $\mathbb{C}^n$ with the usual topology the componentwise conjugation $\theta(z_1,\dots,z_n) = (\overline{z_1},\dots,\overline{z_n})$ is an antilinear involution, not linear, and it is continuous, being the product of $n$ copies of a continuous map. Its fixed subspace is $\mathbb{R}^n$, a closed real subspace, and $\mathbb{C}^n = \mathbb{R}^n \oplus i\mathbb{R}^n$ as a real topological vector space: this is the model real structure, of real dimension $n$, the complex dimension of the space. The same formula with the discrete topology on $\mathbb{C}^n$ is continuous as well, and the example is recorded to fix that the kind of the map and its fixed subspace are data independent of the topology.

Example (products and direct sums, verdict: the componentwise involution is continuous and of the same kind in every factor). Let $(V_i,\theta_i)$ be involutive topological $R$-modules for the same involution $\varsigma$ of $R$, and let $V = \prod_i V_i$ carry the product topology. Then $\theta(x)_i = \theta_i(x_i)$ is a $\varsigma$-semilinear involution of $V$, continuous because each of its components is a composite of the continuous projection with $\theta_i$, and its fixed subspace is $\prod_i V_i^{\theta_i}$. In particular a finite direct sum of involutive topological linear spaces over the same $\varsigma$ is one, with the fixed subspace the direct sum of the fixed subspaces. The scalar involution is the same in every factor, so a linear factor and an antilinear factor cannot be combined: over $\mathbb{C}$ the componentwise map of a complex-linear involution and an antilinear one is not semilinear for a single involution of $\mathbb{C}$, and it is only additive.

The Linear Topology and the Criterion

The Criterion

Theorem (criterion). Let the topology of $V$ be linear, with the submodules $U_0 \supseteq U_1 \supseteq \cdots$ a fundamental system of neighbourhoods of $0$, and let $\theta$ be a $\varsigma$-semilinear involution of $V$. Then $\theta$ is continuous if and only if

$$ \text{for every } n \text{ there is } m \text{ with } \theta(U_m) \subseteq U_n . $$

Proof. If $\theta$ is continuous then $\theta^{-1}(U_n)$ is a neighbourhood of $0$, so it contains some $U_m$, which says $\theta(U_m) \subseteq U_n$. Conversely suppose the condition holds and let $\Omega$ be a neighbourhood of $0$; some $U_n$ is contained in $\Omega$, and for that $n$ the inclusion $\theta(U_m) \subseteq U_n \subseteq \Omega$ shows that $\theta^{-1}(\Omega)$ contains the neighbourhood $U_m$ of $0$, so $\theta$ is continuous at $0$ and hence continuous.

Corollary ($\theta$-stable filtrations). If $\theta(U_n) \subseteq U_n$ for every $n$ then $\theta$ is continuous. For a linear involution the converse holds after a change of filtration: if $T$ is continuous then $U_n' = U_n \cap T(U_n)$ is a submodule with $T(U_n') \subseteq U_n'$, the family $(U_n')$ is a fundamental system of neighbourhoods of $0$ cofinal with $(U_n)$, and the topology is unchanged. For a semilinear involution with $\varsigma \neq \mathrm{id}$ the intersection $U_n \cap \theta(U_n)$ is a subgroup but need not be a submodule, so the filtration is not repaired in this way.

Proof. The first assertion is the criterion with $m = n$. If $T$ is continuous and $T(U_m) \subseteq U_n$, then $U_m' \subseteq T(U_m) \subseteq U_n$, so the family $(U_m')$ is cofinal with $(U_n)$ and defines the same topology, by the change-of-ideal proposition of Topological Modules and Vector Spaces; each $U_n'$ is a submodule because $T$ is linear, the intersection of two submodules, and $T(U_n') = T(U_n) \cap U_n = U_n'$.

Corollary ($I$-adic topologies and the two kinds). Let $I$ be an ideal of $R$, give $V$ the $I$-adic topology with $U_n = I^nV$, and let $\theta$ be a $\varsigma$-semilinear involution of $V$.

(a) If $\theta$ is $R$-linear, that is $\varsigma = \mathrm{id}$, then $\theta(I^nV) \subseteq I^nV$ for every $n$; hence every $R$-linear involution of an $I$-adic topological module is continuous.

(b) If $\varsigma \neq \mathrm{id}$ then $\theta(I^mV) = \varsigma(I)^m\theta(V) \subseteq \varsigma(I)^mV$, so $\theta$ is continuous as soon as $\varsigma(I)^m \subseteq I$ for some $m$; for the coordinatewise involution of the free module $V = R^s$ this condition is also necessary, and then the $I$-adic and the $\varsigma(I)$-adic topologies coincide.

Proof. (a) An element of $I^nV$ is a finite sum $\sum_i r_iv_i$ with $r_i \in I^n$ and $v_i \in V$, and $\theta$ of it is $\sum_i r_i\theta(v_i)$, again in $I^nV$. (b) The displayed inclusion is the semilinearity, and $\varsigma(I)^{mn} \subseteq I^n$ whenever $\varsigma(I)^m \subseteq I$, so the criterion applies with $m$ replaced by $mn$; for $V = R^s$ one has $I^nV = (I^n)^s$ and $\theta(I^nV) = (\varsigma(I)^n)^s$, so continuity forces $\varsigma(I)^m \subseteq I$ on taking $n = 1$; the two topologies coincide by the change-of-ideal proposition.

Remark (what is free and what is not). Part (a) says that over a ring the $I$-adic topology never obstructs a linear involution: continuity restricts the semilinear ones alone, and the criterion (b) is the module form of the $I$-adic criterion of Involutive Topological Rings and Fields, whose necessity direction there is the case $V = R$. For a linear topology that is not $I$-adic the linear case is not free either, and the last example below exhibits a linear involution that is discontinuous for such a topology.

Examples

Example (the $(x)$-adic space $\mathbb{C}[x]$, linear case, verdict: a linear involution, continuous exactly for $b = 0$). Let $V = \mathbb{C}[x]$ with the scalars $\mathbb{C}$ and the linear topology whose neighbourhoods of $0$ are the subspaces $(x)^n$, the polynomials divisible by $x^n$; these are $\mathbb{C}$-subspaces and the topology is Hausdorff because only the zero polynomial is divisible by every $x^n$. For $b \in \mathbb{C}$ let $\varphi_b$ be the substitution $\varphi_b(f)(x) = f(b - x)$. Then $\varphi_b$ is a $\mathbb{C}$-linear map of order two, because the substitution $x \mapsto b - x$ is an involution of the polynomial ring, and it is a linear involution of $V$. Since $\varphi_b((x)^m) = (b-x)^m\mathbb{C}[x]$, the criterion asks whether $(b-x)^m$ is divisible by $x$ for some $m$, and the constant term of $(b-x)^m$ is $(-1)^mb^m$: for $b \neq 0$ it is nonzero, so no $m$ works and $\varphi_b$ is discontinuous, while for $b = 0$ the map is $\varphi_0(f)(x) = f(-x)$ with $\varphi_0((x)^m) = (x)^m$ and $\varphi_0$ is continuous. So in this family the involution is continuous for the single value $b = 0$ and discontinuous for every other value, and the topological category separates involutions that the abstract category cannot see apart. The maps are the ring involutions of Involutive Topological Rings and Fields read here as linear involutions of the topological vector space $\mathbb{C}[x]$.

Example (the $(x)$-adic space $\mathbb{C}[x]$, antilinear case, verdict: an antilinear involution, continuous exactly for $b = 0$). With the same space let $\theta_b(f)(x) = \overline{f(b - x)}$. Then $\theta_b$ is antilinear, since the coefficients are conjugated while the variable is substituted, and $\theta_b^2 = \mathrm{id}$ because the conjugation of the coefficients and the substitution are each applied twice; so $\theta_b$ is a semilinear involution of $V$ for the conjugation of $\mathbb{C}$, and it is not linear. Its continuity is governed by the criterion through the subspaces $\theta_b((x)^m) = \overline{(b-x)^m}\mathbb{C}[x]$, whose generating polynomial has constant term $\overline{(-1)^mb^m}$; exactly as in the linear case, $\theta_b$ is continuous for $b = 0$ and discontinuous for $b \neq 0$. The two examples differ only in the conjugation of the coefficients, hence only in the kind of the map, and the topology decides their continuity by the same computation.

Example (a linear involution discontinuous for a linear topology that is not $I$-adic, verdict: a linear involution, discontinuous). Let $K$ be a field with the discrete topology and let $V = K[x]$ as a $K$-vector space, with the linear topology whose neighbourhoods of $0$ are the subspaces $U_n = \operatorname{span}\{x^{2j} : j \geq n\}$ for $n \geq 1$, together with $U_0 = V$. The intersection of the $U_n$ is $0$, so the topology is Hausdorff, and it is linear; it is not an $I$-adic topology of the field of scalars, a field having no proper nonzero ideal, and it is not the $(x)$-adic topology either, the monomials $x^{2j+1}$ of odd degree being absent from every $U_n$. The map $\theta(f)(x) = f(1 - x)$ is a $K$-linear involution, and for every $m \geq 1$ the subspace $\theta(U_m)$ contains the polynomial $(1 - x)^{2m}$, whose constant term is $1$; no neighbourhood $U_n$ with $n \geq 1$ contains a polynomial with nonzero constant term, so the criterion fails at $n = 1$ and $\theta$ is discontinuous. A diagonal involution in the monomial basis, by contrast, satisfies $\theta(U_n) \subseteq U_n$ and is continuous. So the linear case is free only where the topology is generated by an ideal of the scalars, as in the $I$-adic corollary, and not for a general linear topology.

The Fixed and the Negated Subspaces

Closedness

Theorem (closedness and the fixed module). Let $\theta$ be a topological involution of the Hausdorff topological $R$-module $V$. Then the fixed subspace $V^\theta$ and the negated subspace $V^-$ are closed in $V$; $V^\theta$ is a closed $R^\varsigma$-submodule of $V$ and a topological $R^\varsigma$-module for the subspace topology, and so is $V^-$.

Proof. $V^\theta$ is the equalizer of the continuous maps $\theta$ and $\mathrm{id}_V$, and $V^-$ is the equalizer of $\theta$ and $-\mathrm{id}_V$; the equalizer of two continuous maps into a Hausdorff space is closed, being the preimage of the diagonal. If $x \in V^\theta$ and $r \in R^\varsigma$ then $\theta(rx) = \varsigma(r)\theta(x) = rx$, so $V^\theta$ is an $R^\varsigma$-submodule, and likewise $V^-$; the subring $R^\varsigma$ is closed in $R$ by Involutive Topological Rings and Fields, because $R$ is commutative and $\varsigma$ is a continuous involution, so it is a topological ring in the subspace topology, and the restricted operations of addition and scalar multiplication are continuous.

Corollary (a dense fixed subspace forces the trivial involution). If $V^\theta$ is dense in the Hausdorff space $V$ then $\theta = \mathrm{id}_V$. In particular a continuous involution is determined by its values on any dense subset.

Proof. A dense closed subset is the whole space; $V^\theta = V$ says $\theta = \mathrm{id}$.

The Topological Splitting

Theorem (the averaging maps). Let $\theta$ be a topological involution of $V$ and suppose $2$ is invertible in $R$. Then the averaging maps

$$ \pi_+ = \tfrac12(\mathrm{id}_V + \theta), \qquad \pi_- = \tfrac12(\mathrm{id}_V - \theta), $$

are continuous idempotent $R^\varsigma$-linear endomorphisms of $V$ with $\pi_+ + \pi_- = \mathrm{id}_V$, $\pi_+\pi_- = \pi_-\pi_+ = 0$, images $V^\theta$ and $V^-$ and kernels $V^-$ and $V^\theta$. Consequently

$$ V \cong V^\theta \times V^- $$

as topological $R^\varsigma$-modules, $V$ is the topological direct sum of its fixed and its negated subspace, and $V^\theta$ is a retract of $V$. When $\varsigma = \mathrm{id}$ the maps are $R$-linear and the splitting is a splitting of $R$-modules.

Proof. The maps are composites of the continuous operations of the module with the fixed scalar $1/2$, hence continuous, and $R^\varsigma$-linear because $\pi_\pm(rx) = \tfrac12(rx \pm \varsigma(r)\theta(x)) = r\pi_\pm(x)$ for $r \in R^\varsigma$. They are idempotent with the stated sums, products and kernels, directly from $\theta^2 = \mathrm{id}$ and $2 \cdot \tfrac12 = 1$; the image of $\pi_+$ is $V^\theta$ and the image of $\pi_-$ is $V^-$. Hence $x \mapsto (\pi_+x,\pi_-x)$ is a bijection $V \to V^\theta \times V^-$ with continuous components and continuous inverse the sum, so it is a homeomorphism of topological $R^\varsigma$-modules; the second component of $\pi_+$ being zero exhibits $V^\theta$ as a retract.

Remark. The theorem is the topological refinement of the decomposition of Involutive Linear Spaces, where the type of a linear involution records the dimensions of the two summands. Topologically the dimensions are replaced by the closedness of the summands and the continuity of the projections; the two summands are closed by the theorem above whether or not $2$ is invertible, but the splitting itself needs $2$, exactly as in the abstract theory.

Example (the splitting read on the two $\mathbb{C}[x]$ involutions, verdict: closed summands and a real form). On $V = \mathbb{C}[x]$ with the $(x)$-adic topology and $\varphi_0(f)(x) = f(-x)$, the fixed subspace is the space of even polynomials $\mathbb{C}[x^2]$ and the negated subspace is $x\mathbb{C}[x^2]$, both closed, and $V = \mathbb{C}[x^2] \oplus x\mathbb{C}[x^2]$ is a topological direct sum with continuous projections. On the same space and for the antilinear $\theta_0(f)(x) = \overline{f(-x)}$, the fixed subspace is $\mathbb{R}[x^2] \oplus ix\mathbb{R}[x^2]$, the polynomials whose even coefficients are real and whose odd coefficients are purely imaginary, a closed real subspace, the negated subspace is $i\mathbb{R}[x^2] \oplus x\mathbb{R}[x^2]$, and $V$ is their topological direct sum over the fixed field $\mathbb{R}$ of the conjugation; this is a real form of the complex space. The discontinuous involution $\varphi_1(f)(x) = f(1-x)$ has fixed subspace $\mathbb{C}[x - x^2]$, dense in $V$ and not closed, so the closedness theorem is sharp.

Quotients and Products

Theorem (the quotient carries the involution). Let $W \subseteq V$ be a closed subspace with $\theta(W) \subseteq W$ and let $q : V \to V/W$ be the quotient map. Then $V/W$ is a Hausdorff topological $R$-module, the induced map $\bar\theta(q(x)) = q(\theta(x))$ is a topological involution of $V/W$ of the same kind as $\theta$, and

$$ q(V^\theta) \subseteq (V/W)^{\bar\theta}, $$

with equality when $2$ is invertible in $R$. The quotient fixed subspace can be strictly larger when $2$ is not invertible.

Proof. The quotient of a topological module by a closed submodule is Hausdorff and topological for the quotient topology, by Topological Modules and Vector Spaces, and $\theta$ induces a $\varsigma$-semilinear map of order two because it preserves $W$; it is continuous because $q$ and $\theta$ are. An element of $q(V^\theta)$ has a fixed representative and is fixed, whence the inclusion. If $2$ is invertible and $q(x)$ is fixed, then $\theta(x) - x \in W$; hence $x - \theta x \in W$ and $(x - \theta x)/2 \in V^- \cap W$, because $W$ is a submodule and $2$ is invertible in $R$, while $(x + \theta x)/2 \in V^\theta$; adding the two gives $x \in V^\theta + W$, so $q(x) \in q(V^\theta)$.

Example (strictness when $2$ is not invertible, verdict: a continuous semilinear involution whose quotient fixed subspace is strictly larger). Let $R = \mathbb{Z}[i]$ with the conjugation $\varsigma$, let $V = R$ as a module over itself with the $(2)$-adic topology, and let $\theta = \varsigma$. Then $\theta$ is $\varsigma$-semilinear and not linear, $\theta^2 = \mathrm{id}$, and $\theta(2^nR) = 2^nR$ because $\varsigma(2) = 2$, so $\theta$ is continuous. The fixed submodule $V^\theta = \mathbb{Z}$ has image of two elements in $V/2V$, the ring $\mathbb{Z}[i]/(2)$ of four elements, while the induced involution on the quotient is the identity, because $i \equiv -i \pmod 2$: the quotient fixed subspace has four elements. The inclusion $q(V^\theta) \subsetneq (V/2V)^{\bar\theta}$ is therefore strict, and it is strict exactly because $2$ is not invertible in $\mathbb{Z}[i]$; over a field of characteristic not two the equality above always holds. This is the module-level counterpart of the strictness of Involutive Topological Rings and Fields, which occurs for the same reason.

Remark (no strictness over a field of characteristic not two). For a linear or semilinear involution of a space over a field with $2 \neq 0$ the fixed subspace of the quotient by a stable subspace is exactly the image of the fixed subspace; the strictness is a phenomenon of the coefficient ring, of characteristic two or of a ring in which $2$ is not a unit, and not of the topology.

The Completion

Theorem (extension to the completion). Let $V$ be a Hausdorff topological $R$-module with a linear topology, let $U_0 \supseteq U_1 \supseteq \cdots$ be a fundamental system of submodules of $0$, let $\widehat{V} = \varprojlim_n V/U_n$ be the completion, and let $\theta$ be a topological involution of $V$ of kind $\varsigma$. Then $\theta$ extends to a continuous involution $\widehat{\theta}$ of $\widehat{V}$ of the same kind, the unique continuous extension, and if $2$ is invertible in $R$ then

$$ (\widehat{V})^{\widehat{\theta}} = \overline{\iota(V^\theta)}, $$

the closure being taken in $\widehat{V}$. Consequently the completion adds to the fixed subspace the limits of its elements and nothing else.

Proof. The map $\theta$ is additive and continuous, hence uniformly continuous for the additive uniformity, by the proposition of the first section; the image $\iota(V)$ is dense in $\widehat{V}$ and $\widehat{V}$ is complete Hausdorff, by the completion theorem of Topological Modules and Vector Spaces, so the universal property of the completion, which is the uniform completion of the additive group of Topological Groups, provides a unique continuous additive $\widehat{\theta} : \widehat{V} \to \widehat{V}$ with $\widehat{\theta} \circ \iota = \iota \circ \theta$. Then $\widehat{\theta}^2 \circ \iota = \iota \circ \theta^2 = \iota$, and both $\widehat{\theta}^2$ and the identity are continuous, so $\widehat{\theta}^2 = \mathrm{id}$ by density; the semilinearity $\widehat{\theta}(rx) = \varsigma(r)\widehat{\theta}(x)$ holds on $\iota(V)$ and both sides are continuous in $x$, so it holds on $\widehat{V}$, and for $\varsigma = \mathrm{id}$ the map is linear. For the fixed subspace, $\iota(V^\theta) \subseteq (\widehat{V})^{\widehat{\theta}}$ and the right-hand side is closed, so it contains the closure. Conversely let $a$ be fixed; the image of $\iota$ is dense, so $a$ is the limit of a net $\bigl(\iota(x_k)\bigr)$ with values in $\iota(V)$, and applying $\widehat{\theta}$, which is continuous and fixes $a$, gives $a = \lim \iota(\theta x_k)$ as well; hence $\iota(x_k - \theta x_k) \to 0$ and, multiplying by $\tfrac12$, which exists by hypothesis,

$$ \iota\!\left(\tfrac12(x_k + \theta x_k)\right) = \tfrac12\bigl(\iota(x_k) + \iota(\theta x_k)\bigr) \longrightarrow \tfrac12(a + a) = a, $$

with $\tfrac12(x_k + \theta x_k) \in V^\theta$ for every $k$; so $a$ lies in the closure of $\iota(V^\theta)$.

Corollary (the completed fixed subspace). With the notation of the theorem and $2$ invertible in $R$, the completion of the topological $R^\varsigma$-module $V^\theta$ is the closure of its image in $(\widehat{V})^{\widehat{\theta}}$; that image is the whole fixed subspace exactly when $V^\theta$ is complete.

Example (the completed $\mathbb{C}[x]$, verdict: the two involutions extend and the fixed subspaces are the closures). The $(x)$-adic completion of $\mathbb{C}[x]$ is $\mathbb{C}[[x]]$, and both involutions of the examples above extend: the linear $\varphi_0$ extends to $f(x) \mapsto f(-x)$, whose fixed subspace is the ring of even power series $\mathbb{C}[[x^2]]$, the closure of $\mathbb{C}[x^2]$; the antilinear $\theta_0$ extends to $f(x) \mapsto \overline{f(-x)}$, whose fixed subspace consists of the series whose even coefficients are real and whose odd coefficients are purely imaginary, that is $\mathbb{R}[[x^2]] \oplus ix\mathbb{R}[[x^2]]$, the closure of the fixed subspace of the polynomials. In both cases the completion contributes the limits of the fixed elements and no other fixed element, in agreement with the theorem. The analogues of the discontinuous involutions $\varphi_b$ and $\theta_b$ with $b \neq 0$ are not defined on the completion, exactly because the maps themselves are not continuous.

The Semilinear Case and the Fixed Field

Theorem (the fixed space over the fixed field). Let $F$ be a Hausdorff topological field, $\varsigma$ a continuous involution of $F$, $V$ a Hausdorff topological $F$-vector space and $\theta$ a topological $\varsigma$-semilinear involution of $V$. Then the fixed field $F^\varsigma$ is a closed subfield of $F$, a topological field in the subspace topology, of index two in $F$ when $\varsigma \neq \mathrm{id}$; and $V^\theta$ is a closed topological $F^\varsigma$-vector space. If the characteristic of $F$ is not two then $V = V^\theta \oplus V^-$ as topological $F^\varsigma$-vector spaces.

Proof. That $F^\varsigma$ is a closed subfield, of index two and a topological field when $\varsigma \neq \mathrm{id}$, is the theorem of Involutive Topological Rings and Fields on a topological field with a continuous involution; the statements about $V^\theta$ and the splitting are the closedness theorem and the averaging theorem above, read with $R = F$ and $2$ invertible. That $V^\theta$ is a vector space over $F^\varsigma$, and not merely a module, is the restriction of scalars along the inclusion $F^\varsigma \subseteq F$.

Corollary (the real form). Let $V$ be a complex Hausdorff topological vector space and $\theta$ a continuous antilinear involution of $V$, so that the scalar involution is the conjugation and $F^\varsigma = \mathbb{R}$. Then $V^\theta$ is a closed real subspace and a topological real vector space, the maps $v \mapsto \tfrac12(v + \theta v)$ and $v \mapsto \tfrac12(v - \theta v)$ are continuous real-linear projectors with images $V^\theta$ and $V^-$, and $V = V^\theta \oplus V^-$ as a real topological vector space. When $V$ is finite-dimensional the two real subspaces have real dimension equal to the complex dimension of $V$, so $V^\theta$ is a real form of $V$; this is the topological form of the real structure of Involutive Linear Spaces, and the complexification of Extension of Scalars is its computed instance in the algebra.

Remark (the second sign of the antilinear case). An antilinear map with $\theta^2 = -\mathrm{id}$ is not an involution, its order being four; it has no nonzero fixed vector and the real dimension of a finite-dimensional $V$ carrying it is even. Nothing in this article applies to it, and the algebra it generates is Division Algebras, in Part I; the map is named here only to keep the two signs apart.

Remark (the boundary to the $p$-adic fields). The topological fields with a continuous involution that come after $\mathbb{R}$ and $\mathbb{C}$ are the $p$-adic fields and their finite extensions, where the order-two Galois automorphisms of a quadratic extension give the standard examples; they are Absolute Values, Valuations and Completions and Local Fields, and the finite-dimensional spaces over them are covered by the finite-dimension proposition cited above without any valuation being used here.

What the Topology Adds

The comparison is between the abstract involutive linear space of Involutive Linear Spaces and its topological form. The new datum is continuity, and its consequences are the following.

• Continuity is a hypothesis, and it is free more often than for a ring. It is automatic for the identity, for every linear involution of a finite-dimensional Hausdorff space over a complete valued field, and for every linear involution of an $I$-adic module; it is a genuine restriction for the semilinear involutions, as $\theta_1$ on $\mathbb{C}[x]$ shows, and for the linear ones over a linear topology that is not $I$-adic, as the sparse topology on $K[x]$ shows. Abstractly there is no such question.
• The fixed and the negated subspaces are closed, being equalizers, and the fixed subspace of a semilinear involution is a closed topological module over the closed fixed subring $R^\varsigma$ of the scalars; abstractly the fixed subspace is only a subspace, or over $\mathbb{C}$ only a real subspace. A dense fixed subspace forces the involution to be the identity, and a discontinuous involution may have a fixed subspace that is dense and not closed, as $\mathbb{C}[x - x^2]$ for $\varphi_1$ shows.
• The decomposition is topological. When $2$ is invertible the two averaging maps are continuous projectors, $V$ is the topological direct sum $V^\theta \oplus V^-$ over $R^\varsigma$, and the fixed subspace is a retract; the abstract type $(p,q)$ of a linear involution is thus replaced by a pair of closed complementary summands, and the topological statement holds in infinite dimension, where no type is defined.
• The quotient carries the involution, and over a field of characteristic not two the fixed subspace of the quotient is exactly the image of the fixed subspace; strictness needs $2$ not invertible, as $\mathbb{Z}[i]/(2)$ shows. The contrast with Involutive Topological Rings and Fields, where the strictness of the quotient is the ordinary case, is the effect of the invertibility of $2$ and not of the linear structure.
• The involution extends to the completion, and the fixed subspace of the completion is the closure of the fixed subspace of $V$; the completion is again involutive and of the same kind, and it adds limits of fixed elements and no new fixed element. The discontinuous involutions do not extend at all, which is the sharper statement of the same fact.
• Over $\mathbb{C}$ the antilinear case gives a real form: the fixed subspace is a closed real topological vector space of real dimension the complex dimension, $V$ is the real topological direct sum of it and its image under $i$, and the fixed field of the scalars is the closed subfield $\mathbb{R}$. The linear and the antilinear involution of a complex space are different kinds of object, and the topology keeps them apart: the first has a complex fixed subspace, the second a real one.
• What is not here. The topology restricts the linear structure by continuity and closedness alone; the norms and the operator norms are Normed and Banach Spaces, the topologies on the dual are Duality Theory, the completed tensor products are Topological Tensor Products, and the forms and the adjoints are Hilbert Algebras. Nothing in this article integrates, and no measure is used: the invariant integration is Part III.

Summary

An involutive topological linear space is a topological linear space $V$ over a topological ring $R$ with a continuous $\varsigma$-semilinear involution $\theta$, where $\varsigma$ is a continuous involution of $R$, the involution being linear when $\varsigma = \mathrm{id}$ and antilinear over $\mathbb{C}$ with the conjugation; a topological involution is continuous exactly when it is continuous at $0$ and is then a homeomorphism, uniformly continuous for the additive uniformity. Continuity is free for the identity, for every linear involution of a finite-dimensional Hausdorff space over a complete valued field, and for every linear involution of an $I$-adic module; for a general linear topology with fundamental system of submodules $U_n$ it is equivalent to the condition that for every $n$ there is $m$ with $\theta(U_m) \subseteq U_n$, and for the coordinatewise $\varsigma$-semilinear involution of a free module with the $I$-adic topology it is equivalent to $\varsigma(I)^m \subseteq I$ for some $m$. The involutions $\varphi_b(f)(x) = f(b - x)$ and $\theta_b(f)(x) = \overline{f(b - x)}$ of $\mathbb{C}[x]$ with the $(x)$-adic topology are continuous exactly for $b = 0$, and the second is antilinear while the first is linear; the map $f(x) \mapsto f(1 - x)$ on $K[x]$ with the sparse linear topology is linear and discontinuous.

The fixed subspace $V^\theta$ and the negated subspace $V^-$ of a topological involution of a Hausdorff space are closed, $V^\theta$ is a closed $R^\varsigma$-submodule and a topological module over the closed subring $R^\varsigma$, and a dense fixed subspace forces $\theta = \mathrm{id}$. When $2$ is invertible the averaging maps $\pi_\pm = \tfrac12(\mathrm{id} \pm \theta)$ are continuous idempotent $R^\varsigma$-linear endomorphisms and $V$ is the topological direct sum $V^\theta \oplus V^-$, the fixed subspace being a retract; the discontinuous $\varphi_1$ has fixed subspace $\mathbb{C}[x - x^2]$, dense and not closed, and the antilinear $\theta_0$ has fixed subspace the closed real form $\mathbb{R}[x^2] \oplus ix\mathbb{R}[x^2]$.

A closed $\theta$-stable subspace $W$ gives a Hausdorff quotient with the induced involution, of the same kind; the image of the fixed subspace is contained in the fixed subspace of the quotient, with equality over a field of characteristic not two, while $\mathbb{Z}[i]/(2)$ with the $(2)$-adic topology and the conjugation shows the strictness that occurs when $2$ is not invertible. A topological involution of a Hausdorff module with a linear topology extends uniquely to a continuous involution of the same kind of the completion $\widehat{V} = \varprojlim_n V/U_n$, and when $2$ is invertible the fixed subspace of the extension is the closure of the fixed subspace of $V$; for $\mathbb{C}[x]$ this gives the even power series as the fixed subspace of the extension of $\varphi_0$ and the series with real even and imaginary odd coefficients as that of $\theta_0$. Over a topological field the fixed field $F^\varsigma$ of a continuous involution is a closed subfield, of index two and a topological field when $\varsigma \neq \mathrm{id}$, and the fixed space of a semilinear involution is a closed topological vector space over it; over $\mathbb{C}$ with the conjugation this is the real form, of real dimension the complex dimension.

Summary of Notation

Further Reading

• Nicolas Bourbaki, General Topology, Chapters 1–4 (Springer, 1995), for the uniform spaces, the uniform continuity of continuous homomorphisms and the uniform completion.
• Nicolas Bourbaki, Topological Vector Spaces, Chapters 1–5 (Springer, 1987), for the general theory of topological vector spaces, the linear topologies and the quotients.
• Nicolas Bourbaki, Commutative Algebra, Chapters 1–7 (Springer, 1998), for the $I$-adic topologies, the filtrations by ideals and the completions.
• Jean Dieudonné, Treatise on Analysis, Vol. II (Academic Press, 1970), for topological vector spaces, their subspaces and their quotients.
• Serge Lang, Algebra (Springer, third edition, 2002), for the semilinear maps, the antilinear structures over the complex numbers and the descent of a complex space to a real one.
• Seth Warner, Topological Fields (North-Holland, 1989), for the topological fields, the continuity of an order-two automorphism and the fixed field of a continuous involution.
• Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis I (Springer, second edition, 1979), for the topological algebra background and the involutions of topological algebras, which are Topological Algebras and Banach Algebras and Operator Algebras, later in this part.
• B. J. Pettis, "On continuity and openness of homomorphisms in topological groups", Annals of Mathematics 52 (1950), 293–308, for the automatic continuity of the remark on the closed graph and the Baire-measurable case.