RACSAM. A note on FréchetUrysohn locally convex spaces. 1 Introduction. Jerzy Ka kol and M. López Pellicer


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1 RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 101 (2), 2007, pp Análisis Matemático / Mathematical Analysis A note on FréchetUrysohn locally convex spaces Jerzy Ka ol and M. López Pellicer Abstract. Recently Cascales, Ka ol and Saxon showed that in a large class of locally convex spaces (so called class G) every FréchetUrysohn space is metrizable. Since there exist (under Martin s axiom) nonmetrizable separable FréchetUrysohn spaces C p(x) and only metrizable spaces C p(x) belong to class G, we study another sufficient conditions for FréchetUrysohn locally convex spaces to be metrizable. Una nota sobre espacios de FréchetUrysohn localmente convexos Resumen. Recientemente Cascales, Ka ol y Saxon han probado que en una amplia clase de espacios localmente convexos (llamada clase G) los espacios con la propiedad de FréchetUrysohn son metrizables. Si se admite el axioma de Martin existen espacios C p(x) separables que tienen la propiedad de FréchetUrysohn y que no son metrizables. La metrizabilidad de los espacios C p(x) que pertenecen a la clase G ha motivado el que se estudie en este artículo condiciones suficientes para que los espacios localmente convexos con la propiedad de FréchetUrysohn sean metrizables. 1 Introduction One of the interesting and difficult problems (Malyhin 1978) concerning FréchetUrysohn groups ass if every separable FréchetUrysohn topological group is metrizable [18]; see also [20] and [21] for some counterexamples under various additional settheoretic assumptions. The same question can be formulated in the class of locally convex spaces (lcs). Under Martin s axiom (MA) there exist nonmetrizable analytic (hence separable) FréchetUrysohn spaces C p (X). On the other hand, the Borel Conjecture implies that separable and FréchetUrysohn spaces C p (X) are metrizable. In fact there exist many important classes of lcs for which the FréchetUrysohn property implies metrizablity. We showed in [3, Theorem 2] that (LM) spaces, (DF )spaces (in fact all spaces in class G) are metrizable if and only if they are FréchetUrysohn. In [22, Theorem 5. 7] Webb proved that only finitedimensional Montel (DF )spaces enjoy the Fréchet Urysohn property. We extend this fact by noticing that every FréchetUrysohn hemicompact topological group is a Polish space. The aim of the rest part of the paper is to characterize metrizability of Fréchet Urysohn lcs in terms of certain resolutions. First we prove that for a lcs X its strong dual F is metrizable if and only if F is FréchetUrysohn and X has a bornivorous bounded resolution. This applies to observe that the space of distributions D (Ω) and the space A(Ω) of real analytic functions on an open set Ω R N are not FréchetUrysohn (although they have countable tightness by [3, Corollary 2. 4]). Nevertheless, there exist FréchetUrysohn nonmetrizable lcs which admit a bornivorous bounded resolution, see Example 1. We show however that a FrćhetUrysohn lcs is metrizable if and only if it admits a superresolution. Presentado por Darío Maravall Casesnoves. Recibido: 3 de septiembre de Aceptado: 10 de octubre de Palabras clave / Keywords: Kanalytic space, analytic space, FréchetUrysohn space, topological group. Mathematics Subject Classifications: 46A30 c 2007 Real Academia de Ciencias, España. 127
2 J. Ka ol and M. López Pellicer 2 Notations, Definitions and Elementary Facts A Hausdorff topological space (space) X is said to have a compact resolution if X is covered by an ordered family {A α : α N N }, that is, such that A α A β for α β. Any Kanalytic space has a compact resolution but the converse implication fails in general [19]. For a lcs X a resolution {A α : α N N } is called bounded if each set A α is bounded in X. If additionally every bounded set in X is absorbed by some K α, the resolution {A α : α N N } is called bornivorous. For α = (n ) N N put C n1n 2...n := {Kβ : β = (m l ) l, m j = n j, j = 1,..., }. Clearly K α C n1,...,n for each N. A lcs X is said to have a superresolution if for every finite tuple (n 1,... n p ) of positive integers and every bounded set Q in X there exists α = (m ) N N such that m j = n j for 1 j p and K α absorbs Q. This implies that for any finite tuple (n 1,... n p ) the sequence (C n1,...,n p,n) n is bornivorous in X. Clearly any metrizable lcs X admits a superresolution: For a countable basis (U ) of neighbourhoods of zero in X and α = (n ) N N set K α := n U. Then {K α : α N N } is as required. Moreover, (DF )spaces, regular (LM)spaces admit a bornivorous bounded resolution. Indeed, let X be an (LM)space and let (X n ) n be an increasing sequence of metrizable lcs whose inductive limit is X, see [15] for details. For each n N let (U n) be a countable basis of absolutely convex neihbourhoods of zero in X n such that U n U n+1 and 2U+1 n U n for each N. For each α = (n ) N N set K α := n U n1. Then {K α : α N N } is a bounded resolution. In fact, {K α : α N N } covers X and the set n U n1 is bounded in X n1, hence in the limit space X. If additionally X is regular, i.e. for every bounded set B in X there exists m 1 N such that B is contained and bounded in X m1, then for each N there exists n N such that B n U m1. This yields a sequence α = (m ) N N such that B K α. Any lcs admitting a fundamental sequence (B n ) n of bounded sets has a superresolution; put K α := K n1 for α = (n ) N N. A space X is FréchetUrysohn if for each set A in X and any x A there exists a sequence of elements of A converging to x. A lcs X is barrelled (quasibarrelled) if every closed absolutely convex absorbing (bornivorous) subset of X is a neighbourhood of zero. A lcs X is called Bairelie [17] (bbairelie [16]) if for every increasing (and bornivorous) sequence (A n ) n of absolutely convex closed subsets of E there exists n N such that A n is a neighbourhood of zero in E. Every metrizable (metrizable and barrelled) lcs is bbairelie (Bairelie) and every barrelled bbairelie space is Bairelie. Every FréchetUrysohn lcs is both bbairelie and bornological, [9]. By C p (X) and C c (X) we denote the spaces of realvalued continuous functions on a Tychonov space endowed with the topology of pointwise convergence and the compactopen topology, respectively. 3 Results and Remars Note that for an uncountable compact scattered space X the space C p (X) is FréchetUrysohn [1, Theorem III. 1. 2] nonmetrizable and admits a bounded resolution. (**) Is a FréchetUrysohn lcs metrizable if it admits a strongly bounded resolution? Clearly every lcs with a fundamental sequence of bounded sets generates a bornivorous bounded resolution. In [3] we proved that every FréchetUrysohn lcs in class G is metrizable, so every FéchetUrysohn (DF )space is normable. Nevertheless, we have the following Example 1 Let X be a Σproduct of R I, with uncountable I, formed by all sequences of countable support. Then X contains a FréchetUrysohn nonmetrizable vector subspace having a bornivorous bounded resolution. PROOF. Clearly X is a FréchetUrysohn space, see [13]. Let G be the linear span of the compact set { B := [ 1, 1] I. Set E := F G and denote B F by C. For α = (n ) set K α = n 1 C. Note that Kα : α N N} is a strongly bounded resolution in E. First observe that the family { K α : α N N} is ordered, covers E and each K α is a bounded set in E. Next, let P be a bounded set in E and let A be its closed absolutely convex cover in E. Then A is bounded in R I, hence relatively compact in R I, and 128
3 A note on FréchetUrysohn locally convex spaces since every cluster point of a countable set in X has countable support, A is countably compact in X. But since A is closed in E, we conclude that A is a countably compact subset of E. It is easy to see that A is a Banach dis, i.e. its linear span E A endowed with Minowsi functional norm is a Banach space. Since {nc E A : n N} is a sequence of closed absolutely convex sets in E A covering E A and E A is a Baire space, there is m N such that A mc. So if β = (n ) N N verifies that n 1 = m, then A K β, so that P K β. Therefore the family {K α : α N N } is a strongly bounded resolution in E as stated and E is nonmetrizable since C is not metrizable. In general we note the following Proposition 1 For a lcs X its strong dual (X, β(x, X)) is metrizable if and only if it is FréchetUrysohn and X admits a bornivorous bounded resolution. PROOF. If (X, β(x, X)) is metrizable, then it is FréchetUrysohn and X admits a bornivorous bounded resolution. Now assume that X admits a bornivorous bounded resolution {K α : α N N }. Then the polars Kα of K α in the topological dual X of X form a base of neighbourhoods of zero for the strong topology β(x, X). But the polars Kα in X compose a resolution consisting of equicontinuous sets covering X. This shows that the space (X, β(x, X)) belongs to class G (in sense of Cascales and Orihuela of [2]). If (X, β(x, X)) is FréchetUrysohn, then [3, Theorem 2] yields the metrizability of (X, β(x, X)). Corollary 1 The spaces D (Ω) of distributions and A(Ω) of real analytic functions for an open set Ω R N are not FréchretUrysohn. PROOF. Since D (Ω) is nonmetrizable (quasibarrelled) and is the strong dual of a complete (LF )space D(Ω) of the test functions, we apply Proposition 1. The same argument can be used to the space A(Ω) via [5, Theorem 1. 7 and Proposition 1. 7]. Although (**) has a negative answer we note the following result for a large class of lcs. We need the following somewhat technical Proposition 2 A bbairelie space X is metrizable if and only if X admits a superresolution PROOF. Let {K α : α N N } be a superresolution for X. We may assume that all sets K α are absolutely convex. Then the sets C n1n 2...n (defined above) are also absolutely convex. First observe that for every α = (n ) N N and every neighbourhood of zero U in X there exists N such that C n1,n 2,...,n 2 U. Indeed, otherwise there exists a neighbourhood of zero U in X such that for every N there exists x C n1,n 2,...,n such that x / 2 U. Since x C n1,n 2,...,n for every N, there exists β = (m n) n N N such that x K β, n j = m j, j = 1, 2,...,. Set a n = max { m n : N } for n N and γ = (a n ) n. Since β γ for every N, then K β K γ, so x K γ for all N. Therefore 2 x 0 which provides a contradiction. The claim is proved. Let Y be the completion of X. It is clear that Y is Bairelie. Next, we show that there exists α = (n ) N N such that C n1,n 2,...,n is a neighbourhood of zero in X for each N. Assume that does not exist n 1 N such that C n1 is a neighbourhood of zero. Since (by assumption) the sequence (nc n ) n is bornivorous in X and X is quasibarrelled, we apply [15, ] to deduce that Y = X = n nc n (1 + ɛ) n nc n. But Y is a Bairelie space, so there exists n 1 N such that C n1 is a neighbourhood of zero in Y. Assume that for a finite tuple (n 1,... n p ) of positive integers the set C n1,...,n is a neighbourhood of zero for each 1 p. Since, by assumption, the sequence (nc n1,...,n p,n) n is bornivorous in X, and consequently X = n nc n 1,...,n p,n, we apply the same argument as above to get an integer n p+1 N such that C n1,...,n p,n p+1 is a neighbourhood of zero, which completes the inductive step. This fact combined with the claim provides a countable basis (2 C n1,...,n ) of neighbourhoods of zero, so X is metrizable. 129
4 J. Ka ol and M. López Pellicer The wea topology of any infinitedimensional normed space is nonmetrizable nonbornological but admits a superresolution. Since FréchetUrysohn lcs and spaces C p (X) are bbairelie, Proposition 2 applies to get the following Theorem 1 A FréchetUrysohn lcs is metrizable if and only if it admits a superresolution. C p (X) is metrizable if and only if C p (X) admits a superresolution. As we have already mentioned a FréchetUrysohn lcs in class G is metrizable. But only metrizable spaces C p (X) belong to class G, see [3]. On the other hand, (this fact might be already nown) under (MA)+ (CH) there exist nonmetrizable analytic FréchetUrysohn spaces C p (X). Indeed, by [1, Theorem II.3.2] the space C p (X) is FréchetUrysohn if and only if X has the γproperty, i.e. if for any open cover R of X such that any finite subset of X is contained in a member of R, there exists an infinite subfamily R of R such that any element of X lies in all but finitely many members of R. Gerlits and Nagy [7] showed that under (MA) every subset of reals of cardinality smaller than the continuum has the γproperty. Hence under MA+ (CH) there are uncountable γsubsets Y of reals, see also [8]. Thus for such Y the space C p (Y ) is nonmetrizable separable and FréchetUrysohn. A set of reals A is said to have strong measure zero if for any sequence (t n ) n of positive reals there exists a sequence of intervals (I n ) n covering A with I n < t n for each n N. The Borel Conjecture states that every strong measure zero set is countable; Laver [11] proved that it is relatively consistent with ZFC that the Borel conjecture is true. Assuming the Borel Conjecture, every FréchetUrysohn separable space C p (X) is metrizable. Indeed, since C p (X) is separable and FréchetUrysohn, X admits a weaer separable metric topology ξ and X has the γproperty. Set Y := (X, ξ). Since Y has the γproperty, then C p (Y ) is FréchetUrysohn [7]. By [12, Proposition 4 and Theorem 1] the space Y is zerodimensional. Since Y is metrizable and separable, it is homeomorphic to a subset Z with the γproperty of the Cantor set. On the other hand, GerlitsNagy [7] proved that γsets have strong measure zero. By the Borel Conjecture any γset in the reals is countable. Hence Z is countable. Thus X is countable and C p (X) is metrizable. Since FréchetUrysohn lcs are bbairelie, then every FréchetUrysohn lcs with a fundamental sequence of bounded sets must be a (DF )space. But FréhctUrysohn (DF )spaces are metrizable, [3]. For topological groups the situation is even more striing. The paper [21, Example 1.2] provides nonmetrizable FréchetUrysohn σcompact topological groups but for FréchetUrysohn hemicompact groups X the situation is different. Next proposition extends Webb s theorem [22, Theorem 5.7], who proved that only finitedimensional Montel (DF )spaces enjoy the FréchetUrysohn property. Proposition 3 Every FréchetUrysohn hemicompact topological group X is a locally compact Polish space. PROOF. Let (K n ) n be an increasing sequence of compact sets covering X such that every compact set in X is contained in some K m. Observe first that X is locally compact: It is enough to show that there exists n N such that K n is a neighborhood of the unit of X. Let F be a base of neighborhoods of the unit of X and assume that no K n contains a member of F. For every U F and n N choose x U,n U \ K n, and for each n N let A n = {x U,n : U F}. Since 0 A n for every n N, there exists a sequence (U m,n ) m in F such that x m,n 0, m, where x m,n := x Um,n,n. By [14, Theorem 4] there exists a sequence (n ) of distinct numbers in N and a sequence (m ) in N such that x m,n 0. Since {x m,n : N} is contained in some K p but x m,n / K n, N, we get a contradiction. Hence X is a locally compact FréchetUrysohn group. Since every locally compact FréchetUrysohn topological group is metrizable, see e.g. [10, Theorem 2], we conclude that X is analytic. But any analytic Baire topological group is a Polish space [4, Theorem 5.4], and we reach the conclusion. Acnowledgement. This research was supported for authors by the project MTM of the Spanish Ministry of Education and Science, cofinancied by the European Community (Feder funds). The first named author was also supported by the Grant MNiSW Nr. N N and Ayudas para Estancias en la U. P. V. de Investigadores de prestigio (2007). 130
5 A note on FréchetUrysohn locally convex spaces References [1] Arhangel sii, A. V., (1992). Topological function spaces, Math. and its Appl., Kluwer. [2] Cascales, B. and Orihuela, J., (1987). On Compactness in Locally Convex Spaces, Math. Z., 195, [3] Cascales, B., Ka ol, J. and Saxon, S. A., (2003). Metrizability vs. FréchetUrysohn property, Proc. Amer. Math. Soc., 131, [4] Christensen, J. P. R., (1974). Topology and Borel Structure, North Holland, Amsterdam, Math. Studies 10. [5] Domansi, P. and Vogt, D., (2001). Linear topological properties of the space of analytic functions on the real line, in Recent Progress in Functional Analysis, K. D. Bierstedt et al., eds., NorthHolland Math. Studies 189, [6] Fabian, M., Habala, P., Háje, P., Montesinos, V., Pelant, J. and Zizler, V., (2001). Functional Analysis and InfiniteDimensional Geometry, Canad. Math. Soc., Springer [7] Gerlits, J. and Nagy, Z. S., (1982). Some properties of C(X) I, Top. Appl., 14, [8] Galvin, F. and Miller, A. W., (1984). γsets and other singular sets of real numbers, Top. Appl., 17, [9] Ka ol, J. and Saxon, S., (2003). The FréchetUrysohn property, (LM)spaces and the strongest locally convex topology, Proc. Roy. Irish Acad., 103, 1 8. [10] Ka ol, J., LopezPellicer, M., Peinador, E. and Tarieladze, V., (2007). Lindelöf spaces C(X) over topological groups, Forum Math., DOI: /FORUM.2007.XXX, (to appear). [11] Laver, R.,(1976). On the consistence of Borel s conjecture, Acta Math., 137, [12] McCoy, R. A., (1980). space function spaces, Intern. J. Math., 3, [13] Noble, N., (1970). The continuity of functions on cartesian products, Trans. Math. Soc., 149, [14] Nyios, P. J., (1981). Metrizability and FréchetUrysohn property in topological groups, Proc. Amer. Math. Soc., 83, [15] Pérez Carreras, P. and Bonet, J., (1987). Barrelled Locally Convex Spaces, NorthHolland Math. Studies 131, Notas de Mat., 113, NorthHolland, Amsterdam. [16] Ruess, W., (1976). Generalized inductive limit topologies and barrelledness properties, Pacific J. of Math., 63, [17] Saxon, A., (1972). Nuclear and product spaces, Bairelie spaces, and the strongest locally convex topology, Math. Ann., 197, [18] Shahmatov, D., (2002). Convergence in the presence of algebraic structure, Recent Progress in Topology II, Amsterdam. [19] Talagrand, M., (1979). Espaces de Banach faiblement Kanalytiques, Ann. of Math., 110, [20] Todorčević, S. and Uzcátegui, C., (2005). Analytic spaces, Top. Appl., , [21] Todorčević, S. and Moore, J. T., (2006). The metrization problem for Fréchet groups, Preprint. [22] Webb, J. H.,(1968). Sequential convergence in locally convex spaces, Proc. Cambridge Phil. Soc., 64, Jerzy Ka ol M. López Pellicer Faculty of Mathematics and Informatics Depto. de Matemática Aplicada and IMPA A. Miciewicz University, Universidad Politécnica de Valencia Poznań, Poland E Valencia, Spain 131
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