- Email: [email protected]
The Raĭkov conjecture fails for simple analytical reasons
Journal of Pure and Applied Algebra 216 (2012) 1700–1703
Contents lists available at SciVerse ScienceDirect
Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
The Ra˘ıkov conjecture fails for simple analytical reasons J. Wengenroth Universität Trier, FB IV – Mathematik, 54286 Trier, Germany
article
info
Article history: Received 11 November 2011 Received in revised form 19 December 2011 Available online 27 January 2012 Communicated by B. Keller
abstract We show that a conjecture of Ra˘ıkov from category theory fails for simple reasons which reflect very concrete properties of, for example, partial differential operators. © 2012 Elsevier B.V. All rights reserved.
MSC: 18E05; 18E10; 46A08; 46M15
1. Pre-, semi-, quasi The Ra˘ıkov conjecture states that every semi-abelian category is quasi-abelian. The occurrence of semi and quasi sounds a bit as if this conjecture would be related to the ‘‘theory of piffles’’ invented by A.K. Austin [1]. This, however, is not the case. In many concrete situations, one meets additive categories which are not abelian, for instance if one considers topological abelian groups or vector spaces instead of their naked algebraic counterparts. Nevertheless, in order to apply homological algebra one then indeed needs weaker properties than the usual invertibility of f¯ in the classroom diagram. X
❄ ❄ coim f
f✲ f¯ ✲
Y
✻
∪
im f
In the case of topological abelian groups, f¯ remains bijective and hence a monomorphism and an epimorphism, the noninvertibility is only due to topological reasons. It was thus most natural to define a semi-abelian category just by this property, as was done by Palamodov [9] in his thorough investigation of homological aspects in applications of the theory of locally convex spaces (note that Ra˘ıkov [11] used the term semi-abelian in a different sense). Many homological constructions originally performed with abelian groups then just went through. However, several arguments using pushouts and pullbacks need more information than just the universal properties. One is then led to semi-abelian categories in which cokernels are stable under pullbacks and kernels are stable under pushouts; such categories are called quasi-abelian categories. In detail, stability with respect to pullbacks means the following: if f : X → Y is a cokernel (in the category LCS of all locally convex spaces this means a linear surjection which is continuous and open) and g : Z → Y is any morphism (in LCS, a continuous linear map), then the map h in the pullback diagram f ✲ Y ✻ ✻ g h ✲ P Z
X
E-mail address: [email protected]. 0022-4049/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2012.01.007
J. Wengenroth / Journal of Pure and Applied Algebra 216 (2012) 1700–1703
1701
is again a cokernel (in LCS, P = {(x, z ) ∈ X × Z : f (x) = g (z )} is endowed with the subspace topology of the product and h is the projection onto the second component). A simple but very useful consequence is that for two objects E , F of a quasi-abelian category the following conditions are equivalent. (1) Every short exact sequence 0 → E → Y → F → 0 splits. q
(2) For all exact sequences 0 → E → Y → Z → 0 and morphisms T : F → Z there is a lifting S : F → Y such that T = q ◦ S. f
(3) For all exact sequences 0 → X → Y → F → 0 and morphisms T : X → E there is an extension S : Y → E such that T = S ◦f. This can be used to get ‘‘test sequences’’ which are much easier to deal with than the original sequence (1); see, e.g., [9, theorem 9.1] or [16, section 5.1] for an application of this strategy to Fréchet spaces E. Another way to see quasi-abelianity is that the class of all exact sequences gives an exact structure in the sense of Quillen [10]. This allows us, for example, to define Ext groups without using injective or projective resolutions. Fortunately, many important semi-abelian categories – most prominently, the category LCS – are indeed quasi-abelian. Ra˘ıkov’s conjecture (attributed to him in [6]) was, thus, whether this is always the case. 2. The conjecture fails for very simple reasons Let us briefly recall that the category Bor consists of all bornological locally convex spaces, i.e. every bornivorous set (=absolutely convex set which absorbs all bounded sets) is a neighbourhood of the origin. This class contains all metrizable locally convex spaces and it is stable with respect to locally convex inductive limits and quotients (hence Bor has the same cokernels as LCS) but it is not stable with respect to subspaces. This means that the kernel in Bor of a morphism (=continuous linear map) f : X → Y is the null space of f but endowed with the associated bornological topology having all bornivorous sets as 0-neighbourhoods. Accordingly, the pullback in Bor in the diagram above is the space P with the associated bornological topology. That Bor could be a counterexample to Ra˘ıkov’s conjecture was suspected by Rump, and this was proved in 2005 by Bonet and Dierolf [2] by a sophisticated construction. In 2008, Rump [12] published a different counterexample without mentioning the work of Bonet and Dierolf, which he acknowledged only in an article in 2011 [13]. That paper contains a thorough investigation of the relation between semi-abelian and quasi-abelian categories. In particular, it contains a general criterion for quasi-abelianity which slightly simplifies the arguments showing that Bonet and Dierolf’s construction indeed disproves Ra˘ıkov’s conjecture. In a similar manner, contrived examples in locally convex spaces are used by Sieg and Wegner [14] and Kopylov and Wegner [8] to distinguish other categorical concepts (like left and right semi-abelianity). The aim of this section is to show that one does not need a construction to show that Bor is not quasi-abelian but that this is very easily implied by the existence of non-α -regular inductive limits X = lim Xn of increasing sequences of locally convex
− →
spaces; that is, there is a bounded set in the inductive limit X = n∈N Xn (endowed with the finest locally convex topology making all inclusions Xn ↩→ X continuous) which is not contained in any step Xn . The crucial point in the example of Bonet and Dierolf is the non-β -regularity; i.e., some subset of the first step is bounded in the inductive limit but unbounded in all steps. We refer to Köthe’s book [7, Section 19] and Floret’s survey [5] for more information about inductive limits. For a locally convex inductive limit X = lim Xn , we denote by s : Xn → X the map (xn )n∈N → n∈N xn , which is a − → cokernel in LCS (as well as in Bor, if all Xn ∈ are bornological). Theorem. Let X = lim Xn be an inductive limit of locally convex spaces which is not α -regular. Then there is a normed space Y − → continuously included in X such that the pullback
πY : {(x, y) ∈
Xn × Y : s(x) = y} → Y , (x, y) → y
does not remain open if this space P is endowed with the associated bornological topology. As noted above, the space P endowed with the associated bornological topology is the pullback in Bor (if all Xn are bornological). Then πY is not a cokernel in Bor although s is a cokernel. Therefore, Bor is not quasi-abelian. Proof. Since the inductive limit is not α -regular, we may assume (after omitting some of the steps) that there is a sequence yn ∈ Xn \ Xn−1 which is bounded in X . Let Y be its linear span endowed with the absolutely convex hull B of {yn : n ∈ N} as a unit ball of its norm (note that the sequence (yn )n∈N is linearly independent, which easily implies that the Minkowski functional of B is indeed a norm on Y ). Take any absolutely convex absorbing set V ⊆ Y not absorbing B (for example, the set of all linear combinations c1 y1 + · · · + cm ym with |cn | ≤ 1/n). We claim that M = Xn × V ∩ P is bornivorous. Indeed any bounded set C ⊆ P is contained in A1 × · · · × An × {0} ×· · ·× mB for some n, m ∈ N and bounded sets Ak ⊆ Xk , and hence every (x, y) ∈ C satisfies y = s(x) ∈ Xn . Therefore, πY (C ) is contained in a finite-dimensional subspace of Y , and this implies that πY (C ) is absorbed by V , and hence that C is absorbed by M.
1702
J. Wengenroth / Journal of Pure and Applied Algebra 216 (2012) 1700–1703
We have shown that M is a 0-neighbourhood in P endowed with the associated bornological topology, whereas πY (M ) is not a 0-neighbourhood in Y , since it is contained in V , which itself fails to be a neighbourhood of 0. 3. The conjecture fails for analytical reasons To apply our simple result, one still needs non-α -regular inductive limits of bornological spaces. The first examples of such LB and LF spaces (that is, inductive limits of Banach and Fréchet spaces, respectively) were constructed by Köthe and Grothendieck [7, Section 31.6], and many others followed, for example, in the class of so-called Moscatelli-type spaces as in the construction of Bonet and Dierolf [2]. Such constructed examples might suggest that the failure of quasi-abelianity of Bor is a strange phenomenon at the border of the locally convex theory. Here we want to explain that the opposite is true, since, for many analytical problems, the bornologicity of certain subspaces and α -regularity of inductive limits is at the heart of the matter. To make things concrete, let P (D) : D ′ (Ω ) → D ′ (Ω ) be a linear partial differential operator with constant coefficients on the space of Schwartz distributions over an open set Ω . By the Hahn–Banach theorem, P (D) is surjective if and only if its transposed map P (−D) : D (Ω ) → D (Ω ) is a weak isomorphism onto its image. If the image of P (−D) is closed in D (Ω ), the cokernel is an inductive limit of Fréchet–Schwartz spaces Xn (for this it is enough that the image L is stepwise closed, i.e., L ∩ Xn is closed in Xn for all n; in our concrete situation, however, this is the same as being closed, e.g. by results of Floret [5], where both conditions are characterized by P-convexity for supports). We have the following general result on inductive limits of Fréchet spaces due to the author [15]. Theorem. Let T : E → F be a continuous linear operator between two separated and complete inductive limits of Fréchet– Schwartz spaces with stepwise closed range. The following are equivalent. (1) (2) (2) (3)
The transposed map T t : F ′ → E ′ is surjective. T is a (weak) isomorphism onto its image. imT is a bornological subspace of F . cokT is an (α - or β -) regular (LF) space.
We thus get that the cokernel of P (−D) : D (Ω ) → D (Ω ) is a non-α -regular (LF) space whenever Ω is P-convex for supports and P (D) is not surjective on D ′ (Ω ). 4. The conjecture fails for old reasons The main point in the theorem in Section 2 is that the cokernel P → Y does not remain open under passing to the associated bornological topology. This reflects indeed a general result of Rump [13, dual of Theorem2] about right essential semi-abelian subcategories of a quasi-abelian category. In the concrete situation of Bor ↩→ LCS it states the following stability property. Suppose that Bor is quasi-abelian and let X be a locally convex space which is a topological subspace of some bornological space Z . If T : X → Y is open and Y is bornological, then T : X bor → Y is open. (The proof in this concrete situation is not very difficult: one checks that X is the pullback of the pushout Z × Y /{(x, T (x)) : x ∈ X }.) Having this criterion at hand, we obtain that the Ra˘ıkov conjecture had been solved by Dierolf in [3,4] at about the same time as it was formulated (she thus even solved the Ra˘ıkov problem twice): she proved in those articles that every separated locally convex space Y is the quotient of a complete locally convex space X having finite-dimensional bounded sets. Moreover, her construction shows that, for a separable Banach space Y , say, the space X can be embedded into a product Z of normed spaces Xi , i ∈ I, where the index set has moderate cardinality and thus Z is bornological (in general, a product might be non-bornological if I is of strongly inaccessible cardinality). Since the bounded sets of X are finite dimensional, the associated bornological topology is the finest locally convex topology. If the quotient map X bor → Y were open, Y would also carry the finest locally convex topology, which is not the case if Y is infinite dimensional. References [1] [2] [3] [4] [5]
[6] [7] [8] [9]
A.K. Austin, Miscellanea: modern research in mathematics, Amer. Math. Monthly 84 (7) (1977) 566. MR 1538435. J. Bonet, S. Dierolf, The pullback for bornological and ultrabornological spaces, Note Mat. 25 (1) (2005/06) 63–67. MR 2220453 (2007b:46002). S. Dierolf, Über assoziierte lineare und lokalkonvexe Topologien, Manuscripta Math. 16 (1) (1975) 27–46. MR 0415351 (54 #3439). S. Dierolf, Uber Quotienten vollständiger topologischer Vektorräume, Manuscripta Math. 17 (1) (1975) 73–77. MR 0388022 (52 #8859). K. Floret, Some aspects of the theory of locally convex inductive limits, in: Functional Analysis: Surveys and Recent Results II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979), in: North-Holland Math. Stud., vol. 38, North-Holland, Amsterdam, 1980, pp. 205–237. MR 565407 (81j:46009). V.I. Kuz’minov, A.Ju. Čerevikin, Semiabelian categories, Sibirsk. Mat. Žh. 13 (1972) 1284–1294. 1420. MR 0332926 (48 #11251). G. Köthe, Topological vector spaces. I, in: Translated from the German by D. J. H. Garling. Die Grundlehren der Mathematischen Wissenschaften, Band 159, Springer-Verlag, New York Inc., New York, 1969, MR 0248498 (40 #1750). Ya. Kopylov, S.-A. Wegner, On the notion of a semi-abelian category in the sense of Palamodov, 2011, Preprint. V.P. Palamodov, Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26 (1(157)) (1971) 3–65. MR 0293365 (45 #2442).
J. Wengenroth / Journal of Pure and Applied Algebra 216 (2012) 1700–1703
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[10] D. Quillen, Higher algebraic K -theory. I, in: Algebraic K -theory, I: Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), in: Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129 (49 #2895). [11] D.A. Ra˘ıkov, Semiabelian categories, Dokl. Akad. Nauk SSSR 188 (1969) 1006–1009. MR 0255639 (41 #299). [12] W. Rump, A counterexample to Raikov’s conjecture, Bull. Lond. Math. Soc. 40 (6) (2008) 985–994. MR 2471947 (2009k:18010). [13] W. Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. Pure Appl. Algebra 215 (1) (2011) 44–52. MR 2678698. [14] D. Sieg, S.-A. Wegner, Maximal exact structures on additive categories, 2009, Math. Nachr. (in press). [15] J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Math. 120 (3) (1996) 247–258. MR 1410451 (97m:46006). [16] J. Wengenroth, Derived functors in functional analysis, in: Lecture Notes in Mathematics, vol. 1810, Springer-Verlag, Berlin, 2003, MR 1977923 (2004d:46090).
Contents lists available at SciVerse ScienceDirect
Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
The Ra˘ıkov conjecture fails for simple analytical reasons J. Wengenroth Universität Trier, FB IV – Mathematik, 54286 Trier, Germany
article
info
Article history: Received 11 November 2011 Received in revised form 19 December 2011 Available online 27 January 2012 Communicated by B. Keller
abstract We show that a conjecture of Ra˘ıkov from category theory fails for simple reasons which reflect very concrete properties of, for example, partial differential operators. © 2012 Elsevier B.V. All rights reserved.
MSC: 18E05; 18E10; 46A08; 46M15
1. Pre-, semi-, quasi The Ra˘ıkov conjecture states that every semi-abelian category is quasi-abelian. The occurrence of semi and quasi sounds a bit as if this conjecture would be related to the ‘‘theory of piffles’’ invented by A.K. Austin [1]. This, however, is not the case. In many concrete situations, one meets additive categories which are not abelian, for instance if one considers topological abelian groups or vector spaces instead of their naked algebraic counterparts. Nevertheless, in order to apply homological algebra one then indeed needs weaker properties than the usual invertibility of f¯ in the classroom diagram. X
❄ ❄ coim f
f✲ f¯ ✲
Y
✻
∪
im f
In the case of topological abelian groups, f¯ remains bijective and hence a monomorphism and an epimorphism, the noninvertibility is only due to topological reasons. It was thus most natural to define a semi-abelian category just by this property, as was done by Palamodov [9] in his thorough investigation of homological aspects in applications of the theory of locally convex spaces (note that Ra˘ıkov [11] used the term semi-abelian in a different sense). Many homological constructions originally performed with abelian groups then just went through. However, several arguments using pushouts and pullbacks need more information than just the universal properties. One is then led to semi-abelian categories in which cokernels are stable under pullbacks and kernels are stable under pushouts; such categories are called quasi-abelian categories. In detail, stability with respect to pullbacks means the following: if f : X → Y is a cokernel (in the category LCS of all locally convex spaces this means a linear surjection which is continuous and open) and g : Z → Y is any morphism (in LCS, a continuous linear map), then the map h in the pullback diagram f ✲ Y ✻ ✻ g h ✲ P Z
X
E-mail address: [email protected]. 0022-4049/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2012.01.007
J. Wengenroth / Journal of Pure and Applied Algebra 216 (2012) 1700–1703
1701
is again a cokernel (in LCS, P = {(x, z ) ∈ X × Z : f (x) = g (z )} is endowed with the subspace topology of the product and h is the projection onto the second component). A simple but very useful consequence is that for two objects E , F of a quasi-abelian category the following conditions are equivalent. (1) Every short exact sequence 0 → E → Y → F → 0 splits. q
(2) For all exact sequences 0 → E → Y → Z → 0 and morphisms T : F → Z there is a lifting S : F → Y such that T = q ◦ S. f
(3) For all exact sequences 0 → X → Y → F → 0 and morphisms T : X → E there is an extension S : Y → E such that T = S ◦f. This can be used to get ‘‘test sequences’’ which are much easier to deal with than the original sequence (1); see, e.g., [9, theorem 9.1] or [16, section 5.1] for an application of this strategy to Fréchet spaces E. Another way to see quasi-abelianity is that the class of all exact sequences gives an exact structure in the sense of Quillen [10]. This allows us, for example, to define Ext groups without using injective or projective resolutions. Fortunately, many important semi-abelian categories – most prominently, the category LCS – are indeed quasi-abelian. Ra˘ıkov’s conjecture (attributed to him in [6]) was, thus, whether this is always the case. 2. The conjecture fails for very simple reasons Let us briefly recall that the category Bor consists of all bornological locally convex spaces, i.e. every bornivorous set (=absolutely convex set which absorbs all bounded sets) is a neighbourhood of the origin. This class contains all metrizable locally convex spaces and it is stable with respect to locally convex inductive limits and quotients (hence Bor has the same cokernels as LCS) but it is not stable with respect to subspaces. This means that the kernel in Bor of a morphism (=continuous linear map) f : X → Y is the null space of f but endowed with the associated bornological topology having all bornivorous sets as 0-neighbourhoods. Accordingly, the pullback in Bor in the diagram above is the space P with the associated bornological topology. That Bor could be a counterexample to Ra˘ıkov’s conjecture was suspected by Rump, and this was proved in 2005 by Bonet and Dierolf [2] by a sophisticated construction. In 2008, Rump [12] published a different counterexample without mentioning the work of Bonet and Dierolf, which he acknowledged only in an article in 2011 [13]. That paper contains a thorough investigation of the relation between semi-abelian and quasi-abelian categories. In particular, it contains a general criterion for quasi-abelianity which slightly simplifies the arguments showing that Bonet and Dierolf’s construction indeed disproves Ra˘ıkov’s conjecture. In a similar manner, contrived examples in locally convex spaces are used by Sieg and Wegner [14] and Kopylov and Wegner [8] to distinguish other categorical concepts (like left and right semi-abelianity). The aim of this section is to show that one does not need a construction to show that Bor is not quasi-abelian but that this is very easily implied by the existence of non-α -regular inductive limits X = lim Xn of increasing sequences of locally convex
− →
spaces; that is, there is a bounded set in the inductive limit X = n∈N Xn (endowed with the finest locally convex topology making all inclusions Xn ↩→ X continuous) which is not contained in any step Xn . The crucial point in the example of Bonet and Dierolf is the non-β -regularity; i.e., some subset of the first step is bounded in the inductive limit but unbounded in all steps. We refer to Köthe’s book [7, Section 19] and Floret’s survey [5] for more information about inductive limits. For a locally convex inductive limit X = lim Xn , we denote by s : Xn → X the map (xn )n∈N → n∈N xn , which is a − → cokernel in LCS (as well as in Bor, if all Xn ∈ are bornological). Theorem. Let X = lim Xn be an inductive limit of locally convex spaces which is not α -regular. Then there is a normed space Y − → continuously included in X such that the pullback
πY : {(x, y) ∈
Xn × Y : s(x) = y} → Y , (x, y) → y
does not remain open if this space P is endowed with the associated bornological topology. As noted above, the space P endowed with the associated bornological topology is the pullback in Bor (if all Xn are bornological). Then πY is not a cokernel in Bor although s is a cokernel. Therefore, Bor is not quasi-abelian. Proof. Since the inductive limit is not α -regular, we may assume (after omitting some of the steps) that there is a sequence yn ∈ Xn \ Xn−1 which is bounded in X . Let Y be its linear span endowed with the absolutely convex hull B of {yn : n ∈ N} as a unit ball of its norm (note that the sequence (yn )n∈N is linearly independent, which easily implies that the Minkowski functional of B is indeed a norm on Y ). Take any absolutely convex absorbing set V ⊆ Y not absorbing B (for example, the set of all linear combinations c1 y1 + · · · + cm ym with |cn | ≤ 1/n). We claim that M = Xn × V ∩ P is bornivorous. Indeed any bounded set C ⊆ P is contained in A1 × · · · × An × {0} ×· · ·× mB for some n, m ∈ N and bounded sets Ak ⊆ Xk , and hence every (x, y) ∈ C satisfies y = s(x) ∈ Xn . Therefore, πY (C ) is contained in a finite-dimensional subspace of Y , and this implies that πY (C ) is absorbed by V , and hence that C is absorbed by M.
1702
J. Wengenroth / Journal of Pure and Applied Algebra 216 (2012) 1700–1703
We have shown that M is a 0-neighbourhood in P endowed with the associated bornological topology, whereas πY (M ) is not a 0-neighbourhood in Y , since it is contained in V , which itself fails to be a neighbourhood of 0. 3. The conjecture fails for analytical reasons To apply our simple result, one still needs non-α -regular inductive limits of bornological spaces. The first examples of such LB and LF spaces (that is, inductive limits of Banach and Fréchet spaces, respectively) were constructed by Köthe and Grothendieck [7, Section 31.6], and many others followed, for example, in the class of so-called Moscatelli-type spaces as in the construction of Bonet and Dierolf [2]. Such constructed examples might suggest that the failure of quasi-abelianity of Bor is a strange phenomenon at the border of the locally convex theory. Here we want to explain that the opposite is true, since, for many analytical problems, the bornologicity of certain subspaces and α -regularity of inductive limits is at the heart of the matter. To make things concrete, let P (D) : D ′ (Ω ) → D ′ (Ω ) be a linear partial differential operator with constant coefficients on the space of Schwartz distributions over an open set Ω . By the Hahn–Banach theorem, P (D) is surjective if and only if its transposed map P (−D) : D (Ω ) → D (Ω ) is a weak isomorphism onto its image. If the image of P (−D) is closed in D (Ω ), the cokernel is an inductive limit of Fréchet–Schwartz spaces Xn (for this it is enough that the image L is stepwise closed, i.e., L ∩ Xn is closed in Xn for all n; in our concrete situation, however, this is the same as being closed, e.g. by results of Floret [5], where both conditions are characterized by P-convexity for supports). We have the following general result on inductive limits of Fréchet spaces due to the author [15]. Theorem. Let T : E → F be a continuous linear operator between two separated and complete inductive limits of Fréchet– Schwartz spaces with stepwise closed range. The following are equivalent. (1) (2) (2) (3)
The transposed map T t : F ′ → E ′ is surjective. T is a (weak) isomorphism onto its image. imT is a bornological subspace of F . cokT is an (α - or β -) regular (LF) space.
We thus get that the cokernel of P (−D) : D (Ω ) → D (Ω ) is a non-α -regular (LF) space whenever Ω is P-convex for supports and P (D) is not surjective on D ′ (Ω ). 4. The conjecture fails for old reasons The main point in the theorem in Section 2 is that the cokernel P → Y does not remain open under passing to the associated bornological topology. This reflects indeed a general result of Rump [13, dual of Theorem2] about right essential semi-abelian subcategories of a quasi-abelian category. In the concrete situation of Bor ↩→ LCS it states the following stability property. Suppose that Bor is quasi-abelian and let X be a locally convex space which is a topological subspace of some bornological space Z . If T : X → Y is open and Y is bornological, then T : X bor → Y is open. (The proof in this concrete situation is not very difficult: one checks that X is the pullback of the pushout Z × Y /{(x, T (x)) : x ∈ X }.) Having this criterion at hand, we obtain that the Ra˘ıkov conjecture had been solved by Dierolf in [3,4] at about the same time as it was formulated (she thus even solved the Ra˘ıkov problem twice): she proved in those articles that every separated locally convex space Y is the quotient of a complete locally convex space X having finite-dimensional bounded sets. Moreover, her construction shows that, for a separable Banach space Y , say, the space X can be embedded into a product Z of normed spaces Xi , i ∈ I, where the index set has moderate cardinality and thus Z is bornological (in general, a product might be non-bornological if I is of strongly inaccessible cardinality). Since the bounded sets of X are finite dimensional, the associated bornological topology is the finest locally convex topology. If the quotient map X bor → Y were open, Y would also carry the finest locally convex topology, which is not the case if Y is infinite dimensional. References [1] [2] [3] [4] [5]
[6] [7] [8] [9]
A.K. Austin, Miscellanea: modern research in mathematics, Amer. Math. Monthly 84 (7) (1977) 566. MR 1538435. J. Bonet, S. Dierolf, The pullback for bornological and ultrabornological spaces, Note Mat. 25 (1) (2005/06) 63–67. MR 2220453 (2007b:46002). S. Dierolf, Über assoziierte lineare und lokalkonvexe Topologien, Manuscripta Math. 16 (1) (1975) 27–46. MR 0415351 (54 #3439). S. Dierolf, Uber Quotienten vollständiger topologischer Vektorräume, Manuscripta Math. 17 (1) (1975) 73–77. MR 0388022 (52 #8859). K. Floret, Some aspects of the theory of locally convex inductive limits, in: Functional Analysis: Surveys and Recent Results II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979), in: North-Holland Math. Stud., vol. 38, North-Holland, Amsterdam, 1980, pp. 205–237. MR 565407 (81j:46009). V.I. Kuz’minov, A.Ju. Čerevikin, Semiabelian categories, Sibirsk. Mat. Žh. 13 (1972) 1284–1294. 1420. MR 0332926 (48 #11251). G. Köthe, Topological vector spaces. I, in: Translated from the German by D. J. H. Garling. Die Grundlehren der Mathematischen Wissenschaften, Band 159, Springer-Verlag, New York Inc., New York, 1969, MR 0248498 (40 #1750). Ya. Kopylov, S.-A. Wegner, On the notion of a semi-abelian category in the sense of Palamodov, 2011, Preprint. V.P. Palamodov, Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26 (1(157)) (1971) 3–65. MR 0293365 (45 #2442).
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[10] D. Quillen, Higher algebraic K -theory. I, in: Algebraic K -theory, I: Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), in: Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129 (49 #2895). [11] D.A. Ra˘ıkov, Semiabelian categories, Dokl. Akad. Nauk SSSR 188 (1969) 1006–1009. MR 0255639 (41 #299). [12] W. Rump, A counterexample to Raikov’s conjecture, Bull. Lond. Math. Soc. 40 (6) (2008) 985–994. MR 2471947 (2009k:18010). [13] W. Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. Pure Appl. Algebra 215 (1) (2011) 44–52. MR 2678698. [14] D. Sieg, S.-A. Wegner, Maximal exact structures on additive categories, 2009, Math. Nachr. (in press). [15] J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Math. 120 (3) (1996) 247–258. MR 1410451 (97m:46006). [16] J. Wengenroth, Derived functors in functional analysis, in: Lecture Notes in Mathematics, vol. 1810, Springer-Verlag, Berlin, 2003, MR 1977923 (2004d:46090).