Genericity of Fr\'echet smooth spaces

If a separable Banach space contains an isometric copy of every separable reflexive Fr\'echet smooth Banach space, then it contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach spaces with Fr\'echet smooth dual space. This improves a result of G. Godefroy and N. J. Kalton.Comment: 34 page

On realcompact topological vector spaces

[EN] This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Frechet spaces, (L F)-spaces, and their duals, (D F)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (L F)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no. NN201 2740 33 and for the both authors by the project MTM2008-01502 of the Spanish Ministry of Science and Innovation.Kakol, JM.; López Pellicer, M. (2011). On realcompact topological vector spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 105(1):39-70. https://doi.org/10.1007/s13398-011-0003-0S39701051Argyros S., Mercourakis S.: On weakly Lindelöf Banach spaces. Rocky Mountain J. Math. 23(2), 395–446 (1993). doi: 10.1216/rmjm/1181072569Arkhangel’skii, A. V.: Topological Function Spaces, Mathematics and its Applications, vol. 78, Kluwer, Dordrecht (1992)Batt J., Hiermeyer W.: On compactness in L p (μ, X) in the weak topology and in the topology σ(L p (μ, X), L p (μ,X′)). Math. Z. 182, 409–423 (1983)Baumgartner J.E., van Douwen E.K.: Strong realcompactness and weakly measurable cardinals. Topol. 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Appl. 324, 862–881 (2006). doi: 10.1016/j.jmaa.2005.12.059Ferrando J.C., Ka̧kol J., López-Pellicer M., Saxon S.A.: Quasi-Suslin weak duals. J. Math. Anal. Appl. 339(2), 1253–1263 (2008). doi: 10.1016/j.jmaa.2007.07.081Floret, K.: Weakly compact sets. Lecture Notes in Mathematics, vol. 801, Springer, Berlin (1980)Gillman L., Henriksen M.: Rings of continuous functions in which every finitely generated ideal is principial. Trans. Am. Math. Soc. 82, 366–391 (1956). doi: 10.2307/1993054Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand Reinhold Company, New York (1960)Grothendieck A.: Sur les applications linéaires faiblement compactes d’espaces du type C(K). Can. J. Math. 5, 129–173 (1953)Gullick D., Schmets J.: Separability and semi-norm separability for spaces of bounded continuous functions. Bull. R. Sci. Lige 41, 254–260 (1972)Hager A.W.: Some nearly fine uniform spaces. Proc. Lond. Math. Soc. 28, 517–546 (1974). doi: 10.1112/plms/s3-28.3.517Howes N.R.: On completeness. Pacific J. Math. 38, 431–440 (1971)Isbell, J.R.: Uniform spaces. In: Mathematical Surveys 12, American Mathematical Society, Providence (1964)Ka̧kol J., López-Pellicer M.: Compact coverings for Baire locally convex spaces. J. Math. Anal. Appl. 332, 965–974 (2007). doi: 10.1016/j.jmaa.2006.10.045Ka̧kol, J., López-Pellicer, M.: A characterization of Lindelöf Σ-spaces υ X (preprint)Ka̧kol J., López-Pellicer M., Śliwa W.: Weakly K-analytic spaces and the three-space property for analyticity. J. Math. Anal. Appl. 362(1), 90–99 (2010). doi: 10.1016/j.jmaa.2009.09.026Ka̧kol J., Saxon S.: Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex topology. J. Lond. Math. Soc. 66(2), 388–406 (2002)Ka̧kol J., Saxon S., Todd A.T.: Pseudocompact spaces X and df-spaces C c (X). Proc. Am. Math. Soc. 132, 1703–1712 (2004)Ka̧kol J., Śliwa W.: Strongly Hewitt spaces. Topology Appl. 119(2), 219–227 (2002). doi: 10.1016/S0166-8641(01)00063-3Khan L.A.: Trans-separability in spaces of continuous vector-valued functions. Demonstr. Math. 37, 61–67 (2004)Khan L.A.: Trans-separability in the strict and compact-open topologies. Bull. Korean Math. Soc. 45, 681–687 (2008). doi: 10.4134/BKMS.2008.45.4.681Khurana S.S.: Weakly compactly generated Fréchet spaces. Int. J. Math. Math. Sci. 2(4), 721–724 (1979). doi: 10.1155/S0161171279000557Kirk R.B.: A note on the Mackey topology for (C b (X)*,C b (X)). Pacific J. Math. 45(2), 543–554 (1973)Köthe G.: Topological Vector Spaces I. Springer, Berlin (1969)Kubiś W., Okunev O., Szeptycki P.J.: On some classes of Lindelöf Σ-spaces. Topol. Appl. 153(14), 2574–2590 (2006). doi: 10.1016/j.topol.2005.09.009Künzi H.P.A., Mršević M., Reilly I.L., Vamanamurthy M.K.: Pre-Lindelöf quasi-pseudo-metric and quasi-uniform spaces. Mat. Vesnik 46, 81–87 (1994)Megginson R.: An Introduction to Banach Space Theory. Springer, Berlin (1988)Michael E.: ℵ0-spaces. J. Math. Mech. 15, 983–1002 (1966)Nagami K.: Σ-spaces. Fund. Math. 61, 169–192 (1969)Narayanaswami P.P., Saxon S.A.: (LF)-spaces, quasi-Baire spaces and the strongest locally convex topology. Math. Ann. 274, 627–641 (1986). doi: 10.1007/BF01458598Negrepontis S.: Absolute Baire sets. Proc. Am. Math. Soc. 18(4), 691–694 (1967). doi: 10.2307/2035440Orihuela J.: Pointwise compactness in spaces of continuous functions. J. Lond. Math. Soc. 36(2), 143–152 (1987). doi: 10.1112/jlms/s2-36.1.143Orihuela, J.: On weakly Lindelöf Banach spaces. In: Bierstedt, K.D. et al. (eds.) Progress in Functional Analysis, pp. 279–291. Elsvier, Amsterdam (1992). doi: 10.1016/S0304-0208(08)70326-8Orihuela J., Schachermayer W., Valdivia M.: Every Readom–Nikodym Corson compact space is Eberlein compact. Stud. Math. 98, 157–174 (1992)Orihuela, J., Valdivia, M.: Projective generators and resolutions of identity in Banach spaces. Rev. Mat. Complut. 2(Supplementary Issue), 179–199 (1989)Pérez Carreras P., Bonet J.: Barrelled Locally Convex Spaces, Mathematics Studies 131. North-Holland, Amsterdam (1987)Pfister H.H.: Bemerkungen zum Satz über die separabilität der Fréchet-Montel Raüme. Arch. Math. (Basel) 27, 86–92 (1976). doi: 10.1007/BF01224645Robertson N.: The metrisability of precompact sets. Bull. Aust. Math. Soc. 43(1), 131–135 (1991). doi: 10.1017/S0004972700028847Rogers C.A., Jayne J.E., Dellacherie C., Topsøe F., Hoffman-Jørgensen J., Martin D.A., Kechris A.S., Stone A.H.: Analytic Sets. Academic Press, London (1980)Saxon S.A.: Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology. Math. Ann. 197(2), 87–106 (1972). doi: 10.1007/BF01419586Schawartz L.: Radom Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press, Oxford (1973)Schlüchtermann G., Wheller R.F.: On strongly WCG Banach spaces. Math. Z. 199(3), 387–398 (1988). doi: 10.1007/BF01159786Schlüchtermann G., Wheller R.F.: The Mackey dual of a Banach space. Note Math. 11, 273–287 (1991)Schmets, J.: Espaces de functions continues. Lecture Notes in Mathematics, vol 519, Springer-Verlag, Berlin-New York (1976)Talagrand M.: Sur une conjecture de H. H. Corson. Bull. Soc. Math. 99, 211–212 (1975)Talagrand M.: Espaces de Banach faiblement K-analytiques. Ann. Math. 110, 407–438 (1979)Talagrand M.: Weak Cauchy sequences in L 1(E). Am. J. Math. 106(3), 703–724 (1984). doi: 10.2307/2374292Tkachuk V.V.: A space C p (X) is dominated by irrationals if and only if it is K-analytic. Acta Math. Hungar. 107(4), 253–265 (2005)Tkachuk V.V.: Lindelöf Σ-spaces: an omnipresent class. RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat. 104(2), 221–244 (2010). doi: 10.5052/RACSAM.2010.15Todd A.R., Render H.: Continuous function spaces, (db)-spaces and strongly Hewitt spaces. Topol. 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Springer, Berlin (1974

On nonseparable banach spaces

Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pelczynski that states that the isomorphic embeddability of L1(— 1, 1)α into X* implies, for any infinite cardinal a, the isomorphic embedding of /J into X, gets the following form: if α = ω, has been proved from Pelczynski; if α > ω+, the proof is given in this paper; if α = ω+, in Z.F.C. + C.H., an example discovered by Haydon gives a negative answer; if α = ω+, in Z.F.C.+ C.H. + M.A., is also proved in this paper. © 1982 American Mathematical Society

Examples concerning heredity problems of WCG banach spaces

We present two examples of WCG spaces that are not hereditarily WCG. The first is a space with an unconditional basis, and the second is a space X such that X** is WCG and X** does not contain l1. The non-WCG subspace Y of X has the additional property that Y** is not WCG and X/Y is reflexive. © 2004 American Mathematical Society

A proof of Halpern-Läuchli partition theorem

A proof of the Halpern Läuchli partition theorem and its version for strong subtrees is given. We prove a general statement which has, as an immediate consequence, the above-mentioned results. The proof of this is direct and avoids metamathematical arguments. Some consequences for partitions of finite products of metric spaces are also presented. © 2002 Academic Press

Chapter 23 Descriptive set theory and Banach spaces

The goal of this chapter is to reveal the power of descriptive set theory in penetrating the structure of Banach spaces. The chapter is divided into three subchapters, each with its own introduction. Subchapters one, two and three were mostly written by the second, third, and first authors, respectively. Space limitations forced us to leave out many fundamental results and on-going research in this exciting interface. We do hope, however, that our article gives a strong flavor of this aspect of Banach space theory. © 2003 Elsevier B.V. All rights reserved

Reznichenko families of trees and their applications

Examples of Talagrand, Gul'ko and Corson compacta resulting from Reznichenko families of trees are presented. The Kσ δ property for weakly K-analytic Banach spaces with an unconditional basis is proved. © 2008 Elsevier Inc. All rights reserved

Talagrand's Kσ δ problem

We show that the Banach spaces C (K) with K either an adequate Talagrand compact or a quasi adequate σ-Eberlein Talagrand compact are Kσ δ subsets of their second dual endowed with the weak* topology. As consequence we obtain that weakly K-analytic Banach spaces with an unconditional basis are Kσ δ. We also provide an example of a Talagrand compact K such that C (K) is not Kσ δ in its second dual. This answers a problem posed by M. Talagrand. © 2008 Elsevier B.V. All rights reserved