We show by example that the associative law does not hold for tensor products
in the category of general (not necessarily locally convex) topological vector
spaces. The same pathology occurs for tensor products of Hausdorff abelian
topological groups.Comment: 7 pages; v2: new version focusses on the real cas

Glockner, Helge

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summary:We show by example that the associative law does not hold for tensor products in the category of general (not necessarily locally convex) topological vector spaces. The same pathology occurs for tensor products of Hausdorff abelian topological groups

Glöckner, Helge

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Tensor products in the category of topological vector spaces are not associative

Helge Glöckner

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Commentationes Mathematicae Universitatis CarolinaeHelge GlöcknerTensor products in the category of topological vector spaces are not associativeCommentationes Mathematicae Universitatis Carolinae, Vol. 45 (2004), No. 4, 607--614Persistent URL: http://dml.cz/dmlcz/119489Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2004Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 45,4 (2004)607–614 607Tensor products in the categoryof topological vector spaces are not associativeHelge GlöcknerAbstract. We show by example that the associative law does not hold for tensor productsin the category of general (not necessarily locally convex) topological vector spaces. Thesame pathology occurs for tensor products of Hausdorff abelian topological groups.Keywords: topological tensor product, associative law, non-locally convex spaceClassification: 46A16, 46A32; 22A05IntroductionLet A be a class of (not necessarily Hausdorff) real topological vector spaces(resp., of abelian topological groups), such that A is closed under the formationof finite cartesian products. Given E1, . . . , En ∈ A (where 2 ≤ n ∈ N), we callan element T ∈ A, together with an n-linear (resp., n-additive) continuous mapτ : E1 × · · · ×En → T,a tensor product of E1, . . . , En in the class A if for every E ∈ A and n-linear(resp., n-additive) continuous map f : E1 × · · · × En → E, there exists a uniquecontinuous linear map (resp., continuous homomorphism) f̃ : T → E such thatf̃ ◦ τ = f .For example, the tensor products in the class of Hausdorff locally convex spacesare the projective tensor products , going back to Grothendieck’s memoir [8]. Inthis case, an explicit description of the locally convex topology (by means ofsuitable cross-seminorms) is available, and it is well-known that an associative lawholds for iterated projective tensor products; this is important for applications intopological algebra.Two-fold tensor products E ⊗ F in the class of real topological vector spaceshave been studied in [15], [17], [7], [10], [5] and the breakthrough papers [13],[14]. For two-fold tensor products in various classes of abelian topological groups,see [9], [6], [15], [1], [11], and [12]. In none of these works, higher tensor prod-ucts or iterated tensor products are discussed, and accordingly the question ofassociativity of tensor products has not been raised there.608 H.GlöcknerThe present paper intends to close this gap. Based on an explicit descriptionof the topology on tensor products in the category of real topological vectorspaces provided in Section 1 (Proposition 1), we first describe a sufficient conditionensuring that (E1 ⊗ E2) ⊗ E3 be canonically isomorphic to E1 ⊗ (E2 ⊗ E3): itsuffices that the outer factors E1 and E3 be locally bounded (Proposition 2).We then establish the main result: For E := RN, none of the tensor products(E ⊗E)⊗E, E ⊗ (E ⊗E) and E ⊗E ⊗E (in the class of real topological vectorspaces) are naturally isomorphic (Theorem 1). Likewise, the associative law failsfor tensor products in the categories of Hausdorff real topological vector spacesand Hausdorff abelian topological groups (Remark 1).1. Description of the topology on tensor productsIn the following, topological vector spaces are not presumed Hausdorff, nor aretopological groups, unless we explicitly say the contrary.Proposition 1. Given real topological vector spaces E1, . . . , En, we let T :=E1⊗R · · ·⊗REn be the (algebraic) tensor product of the vector spaces E1, . . . , En,and τ : E1× · · · ×En → T , τ(x1, . . . , xn) := x1⊗ · · · ⊗ xn. Let B be the set of allsubsets U of T of the form(1) U =∑k∈Nτ(Uk,1 × · · · × Uk,n) :=⋃k∈Nk∑ℓ=1τ(Uℓ,1 × · · · × Uℓ,n),where (Uk,j)k∈N is a sequence of balanced zero-neighbourhoods in Ej , for j =1, . . . , n. Then the following holds.(a) B is a basis for the filter of zero-neighbourhoods of some topology O on Tmaking T a real topological vector space.(b) τ : E1 × · · · ×En → (T,O) is continuous.(c) (T,O), together with the continuous n-linear map τ , is a tensor productof E1, . . . , En in the category of real topological vector spaces.(d) If E1, . . . , En are Hausdorff, then T is Hausdorff and hence is the tensorproduct of E1, . . . , En in the category of Hausdorff real topological vectorspaces.(e) If E1, . . . , En are Hausdorff, then (T,O), considered as an abelian topo-logical group, together with the continuous n-additive map τ , is a tensorproduct of the Hausdorff abelian topological groups (E1,+), . . . , (En,+)in the category of Hausdorff abelian topological groups.Proof: (a) It is obvious that every U ∈ B is balanced, absorbing, and thattU ∈ B for any t ∈ R×; hence conditions (EVI) and (EVII) of [3, I, §1, No. 5,Proposition 4] are satisfied. In order that B be a basis of a vector topology, itremains to verify condition (EVIII). To this end, let U be as in (1). We findTensor products in the category of topological vector spaces are not associative 609balanced zero-neighbourhoods Vk,j in Ej for each k ∈ N and j ∈ {1, . . . , n}such that Vk,j ⊆ U2k−1,j ∩ U2k,j. Then, re-ordering terms and abbreviatingVk,1 ⊗ · · · ⊗ Vk,n := τ(Vk,1 × · · · × Vk,n), we find that(∑k∈NVk,1 ⊗ · · · ⊗ Vk,n)+(∑k∈NVk,1 ⊗ · · · ⊗ Vk,n)= V1,1⊗· · · ⊗ V1,n + V1,1⊗ · · · ⊗V1,n + V2,1⊗ · · · ⊗V2,n + V2,1⊗ · · · ⊗V2,n + · · ·⊆∑k∈NUk,1 ⊗ · · · ⊗ Uk,n .Thus B is a basis for the filter of zero-neighbourhoods of a vector topology Oon T .(b) In order that the n-linear map τ be continuous, we only need to show it iscontinuous at zero (see [3, Chapter I, §1, No. 6, Proposition 5]). Now, given anyzero neighbourhood U ∈ B as in (1), we have τ−1(U) ⊇ U1,1 × · · · × U1,n, whichis a neighbourhood of (0, . . . , 0).(c) Suppose that f : E1 × · · · × En → E is a continuous n-linear map to areal topological vector space E. Since (T, τ) is the (algebraic) tensor product ofthe vector spaces E1, . . . , En, there is a unique linear map f̃ : T → E such thatf̃ ◦ τ = f . It only remains to show that f̃ is continuous. To this end, let W ⊆ Ebe a zero-neighbourhood. Then standard arguments provide a sequence of zero-neighbourhoods Wk ⊆ E such that∑k∈NWk ⊆ W . The n-linear map f beingcontinuous, for each k ∈ N we find balanced zero-neighbourhoods Uk,j ⊆ Ejfor j = 1, . . . , n such that f(Uk,1 × · · · × Uk,n) ⊆ Wk. Then the set U :=∑k∈N τ(Uk,1 × · · · × Uk,n) ∈ B is a zero-neighbourhood in T , andf̃(U) =∑k∈Nf̃(τ(Uk,1 × · · · × Uk,n))=∑k∈Nf(Uk,1 × · · · × Uk,n) ⊆∑k∈NWk ⊆W.Thus the linear map f̃ is continuous at zero and thus continuous.(d) The proof is by induction. For n = 2, the assertion is Turpin’s celebratedresult (see [13], [14]). Now, by induction E1 ⊗ · · · ⊗ En−1 is Hausdorff, andhence so is F := (E1 ⊗ · · · ⊗ En−1) ⊗ En. The continuous n-linear map f :E1 × · · · ×En → F , (x1, . . . , xn) 7→ (x1 ⊗ · · · ⊗ xn−1)⊗ xn induces a continuouslinear map f̃ : T = E1⊗· · ·⊗En → F , determined by f̃ ◦ τ = f . It is known fromabstract algebra that f̃ is an isomorphism of vector spaces. Since F is Hausdorffand f̃ is a continuous injection, T is Hausdorff.(e) To outline the idea, let us assume that n = 2 (the general case beinganalogous). Suppose that f : E1 × E2 → A is a continuous bi-additive map toan abelian topological group A. Using the given scalar multiplication also on the610 H.Glöcknerright, we consider E1 as a (Z,R)-bimodule. Similarly, E2 is considered as an(R,Z)-bimodule. Then f is R-balanced in the sense of [4, p. 161]. In fact, forany x ∈ E1, y ∈ E2 and q ∈ Q, say q =mn with m ∈ Z and n ∈ N \ {0}, wehave f(qx, y) = f(m 1nx, n1ny)= f(n 1nx,m1ny)= f(x, qy), whence f(rx, y) =f(x, ry) for all r ∈ R, by continuity. Now, f being R-balanced, there is a uniquelydetermined homomorphism of groups f̃ : T → A such that f̃ ◦ τ = f (cf. [4,pp. 161–162]). As in (c), we see that f̃ is continuous. 2. A criterion for associativity of tensor productsGiven real topological vector spaces E1, E2, E3, there is a continuous linearmap φ : E1 ⊗E2 ⊗E3 → (E1 ⊗E2)⊗E3, uniquely determined by φ(x⊗ y⊗ z) =(x⊗y)⊗z, and φ is an isomorphism of vector spaces (cf. proof of Proposition 1(d)).Likewise, there is a unique continuous linear map (and isomorphism of vectorspaces) ψ : E1⊗E2⊗E3 → E1⊗(E2⊗E3), determined by ψ(x⊗y⊗z) = x⊗(y⊗z).The following proposition describes criteria ensuring that φ, ψ and θ := ψ−1◦φbe isomorphisms of topological vector spaces. Since φ and ψ are isomorphismsof vector spaces, we can always identify (E1 ⊗ E2) ⊗ E3, E1 ⊗ (E2 ⊗ E3) andE1 ⊗E2 ⊗E3 as vector spaces for simplicity of notation; only the topologies maydiffer.Proposition 2. In the preceding situation, we have:(a) if E3 is locally bounded, then φ is an isomorphism of topological vectorspaces;(b) if E1 and E3 are locally bounded, then the natural isomorphism of vectorspacesθ : (E1 ⊗ E2)⊗ E3 → E1 ⊗ (E2 ⊗ E3)taking (x ⊗ y) ⊗ z to x ⊗ (y ⊗ z) is an isomorphism of topological vectorspaces.Proof: (a) Let B be a bounded, balanced zero-neighbourhood in E3. Then{tB : t ∈ R×} is a basis of zero-neighbourhoods in E3. Hence a basis of zero-neighbourhoods of E1 ⊗ E2 ⊗ E3 is given by the sets of the form∑n∈NUn ⊗ Vn ⊗ (tnB) =∑n∈N(tnUn)⊗ Vn ⊗B,where Un and Vn are open zero-neighbourhoods in E1 and E2, respectively, andtn ∈ R×. Replacing Un with tnUn, we see that it suffices to take tn = 1 for alln ∈ N here: the setsW :=∑n∈NUn ⊗ Vn ⊗BTensor products in the category of topological vector spaces are not associative 611form a basis of zero-neighbourhoods. Suppose such a W is given. We choose abijection λ : N → N2 and define Pλ(n) := Un and Qλ(n) := Vn for n ∈ N. ThenW =∑n∈NPλ(n) ⊗Qλ(n) ⊗B=∑(j,k)∈N2P(j,k) ⊗Q(j,k) ⊗B ⊇∑j∈N(∑k∈NP(j,k) ⊗Q(j,k))⊗B,where the right hand side is a zero-neighbourhood in (E1 ⊗ E2)⊗ E3. Thus φ isopen and thus φ is an isomorphism of topological vector spaces.(b) If also E1 is locally bounded, we see in the same way that ψ is an isomor-phism of topological vector spaces, whence so is θ = ψ ◦ φ−1. 3. Examples where associativity failsWe show that tensoring in the category of topological vector spaces is notassociative.Theorem 1. Let E := RN with the product topology. Then the following holds.(a) The canonical isomorphism of vector spacesθ : (E ⊗ E)⊗ E → E ⊗ (E ⊗ E)is not continuous (and hence not an isomorphism of topological vectorspaces).(b) The canonical isomorphisms of vector spacesφ : E ⊗ E ⊗ E → (E ⊗ E)⊗ E and ψ : E ⊗ E ⊗ E → E ⊗ (E ⊗ E)are continuous but not open.Proof: (a) The set Un,ε := {z ∈ R : |z| < ε}n × R{n+1,n+2,...} is a zero-neighbourhood in E, for each ε > 0 and n ∈ N. HenceU :=∑n∈NU2n,2−n ⊗(∑k∈N(U2n,2−k ⊗ U2n,1))is a zero-neighbourhood in E ⊗ (E ⊗E). Then U is not a zero-neighbourhood in(E ⊗E)⊗ E. In fact, otherwise we find zero-neighbourhoods An,k, Bn,k and Cnin E such thatV :=∑n∈N(∑k∈NAn,k ⊗Bn,k)⊗ Cn ⊆ U.612 H.GlöcknerThere is N ∈ 2N such that R eN ⊆ C1, where eN := δN,• : N → {0, 1} ⊆ R isdefined using Kronecker’s delta. The sets A1,k and B1,k being absorbing, we thenhave(2)(∑k∈NE ⊗ E)⊗ R eN ⊆ V ⊆ U.Let p : E ⊗ E ⊗ E → RN ⊗ RN be the linear map uniquely determined byp(x⊗ y ⊗ z) = z(N) ·(x|{1,... ,N})⊗(y|{1,... ,N}).We now identify RN with RN × {0} ⊆ RN. Given v ∈ RN ⊗ RN and R ∈ R×,we have (Rv) ⊗ eN ∈ U by (2), whence (Rv) ⊗ eN =∑k,n∈N xn ⊗ yn,k ⊗ zn,kfor suitable xn ∈ U2n,2−n , yn,k ∈ U2n,2−k , and zn,k ∈ U2n,1 (depending on R),almost all of which are zero. Hencev = p(v ⊗ eN ) =∑k,n∈Ncn,kR · un ⊗ vn,k =∑n∈Nun ⊗(∑k∈Ncn,kR · vn,k)︸ ︷︷ ︸=:wnwith un := xn|{1,... ,N}, vn,k := yn,k|{1,... ,N} and cn,k := zn,k(N). Thus(3) v =N/2∑n=1un ⊗ wn + ρwith ρ :=∑n>N/2 un ⊗ wn =∑Ni,j=1(∑n>N/2∑k∈Ncn,kR un(i)vn,k(j))ei ⊗ ej .Note that, as 2n > N ≥ i, j, we have |un(i)| < 2−n, |vn,k| < 2−k and |cn,k| < 1.Thus∑n>N/2∑k∈N∣∣ cn,kR un(i)vn,k(j)∣∣ ≤ 1|R|∞∑n,k=12−n2−k = 1|R|,which can be made arbitrarily small for large |R|. Thus (3) shows that v iscontained in the closure of the set S of N/2-fold sums of elementary tensors, withrespect to the canonical Hausdorff vector topology (∼= RN2) on RN⊗RN . Hence Sis dense in RN ⊗RN . However, under the usual isomorphism RN ⊗RN ∼=MN (R)the set S corresponds to the set of matrices of rank ≤ N/2, which is not dense inMN (R) as it does not meet the open set GLN (R) of invertible matrices. We havereached a contradiction.(b) If one of φ and ψ were an isomorphism of topological vector spaces, then alsothe other, because ψ = α◦φ◦β, where the linear maps β : E⊗E⊗E → E⊗E⊗Eand α : (E ⊗E)⊗E → E ⊗ (E ⊗E) determined by β(x⊗ y⊗ z) := y⊗ z⊗ x andα((x⊗y)⊗z) = z⊗(x⊗y) are isomorphisms of topological vector spaces. Thus θ =ψ ◦ φ−1 would be an isomorphism of topological vector spaces, contradicting (a).Tensor products in the category of topological vector spaces are not associative 613Remark 1. The spaces E ⊗ E, (E ⊗ E) ⊗ E and E ⊗ (E ⊗ E) are also therespective tensor products in the category of Hausdorff real topological vectorspaces (Proposition 1(d)), whence Theorem 1(a) entails that tensoring is notassociative in this (more interesting) category. Nor is tensoring associative inthe category of complete Hausdorff real topological vector spaces, since E ⊗ E,(E⊗E)⊗E and E⊗(E⊗E) are complete (see [14]). Likewise, considering E⊗E,(E ⊗ E) ⊗ E and E ⊗ (E ⊗ E) as tensor products in the category of Hausdorffabelian topological groups (Proposition 1(e)), we deduce that tensoring is notassociative in the category of Hausdorff abelian topological groups.Remark 2. Note that the definition of tensor products and all of the resultsobtained (except for Proposition 1(d), (e) and Remark 1) remain meaningful andcorrect when R is replaced with an arbitrary complete valued field K (for instance,the field of p-adic numbers). This observation is of interest in connection with[2], where a framework of differential calculus over non-discrete topological fieldsis described. In this context, it is desirable to know precisely which constructionsof topological algebra work over general topological fields (or at least completevalued fields), in contrast to those which depend on specific properties of the realnumber field, or on local convexity.Open problems. Does Proposition 1(d) carry over to Hausdorff topological vec-tor spaces over complete valued fields K other than R (or C) ? Are tensor productsof complete Hausdorff topological K-vector spaces in the class of Hausdorff topo-logical K-vector spaces always complete, as in the real case (discussed in [14]) ?Acknowledgements. The author wishes to thank K.-H. Neeb for a suggestionwhich lead to Proposition 1(e).References[1] Akmalov N.A., Topologies in the tensor product of abelian groups (in Russian), Dokl. Akad.Nauk UzSSR (1980), no. 1, 3–6.[2] Bertram W., Glöckner H., Neeb K.-H., Differential calculus over general base fields andrings, Exp. Math. 22 (2004), 213–282.[3] Bourbaki N., Topological Vector Spaces, Chapters 1–5, Springer, Berlin, 1987.[4] Cohn P.M., Algebra, Vol. II, 2nd Ed., John Wiley, Chichester, 1989.[5] Despland J.C., Produit tensoriels de deux éspaces de Banach, Rev. Roumaine Math. PuresAppl. 27 (1982), 25–32.[6] Garling D.J.H., Tensor products of topological Abelian groups, J. Reine Angew. Math. 223(1966), 164–182.[7] Iyahen S.O., On certain classes of linear topological spaces II, J. London Math. Soc. 3(1971), 609–617.[8] Grothendieck A., Produit tensoriels topologiques et éspaces nucléaires, Mem. Amer. Math.Soc. 1955 (1955), no. 16, 140 pp.[9] Hofmann K.H., Tensorprodukte lokal kompakter topologischer Gruppen, J. Reine Angew.Math. 216 (1964), 134–149.[10] Hollstein R., Über die Metrisierbarkeit von projektiven Tensorprodukten, ManuscriptaMath. 9 (1973), 201–209.614 H.Glöckner[11] Loth P., Topologically pure extensions, pp. 191–201 in: “Abelian Groups, Rings and Mod-ules” (Perth, 2000), Contemp. Math. 273, AMS, Providence, 2001.[12] Sahleh H., Tensor product of abelian topological groups and 2nd nilpotent, Int. Math. J. 2(2002), no. 11, 1081–1087.[13] Turpin Ph., Sur certains produits tensoriels topologiques, C.R. Acad. Sci. Paris, Sér. A 291(1980), 405–407.[14] Turpin Ph., Produits tensoriels d’espaces vectoriels topologiques, Bull. Soc. Math. Fr. 110(1982), 3–13.[15] Tomášek S., On tensor products of Abelian groups, Comment. Math. Univ. Carolinae 6(1965), 73–83.[16] Tomášek S., Some remarks on tensor products, ibid. 6 (1965), 85–96.[17] Waelbroeck L., The tensor product of a locally pseudo-convex and a nuclear space, StudiaMath. 38 (1970), 101–104.TU Darmstadt, FB Mathematik AG 5, Schlossgartenstr. 7, 64289 Darmstadt,GermanyE-mail : gloeckner@mathematik.tu-darmstadt.de(Received August 1, 2003)

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Tensor products in the category of topological vector spaces are not associative

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