Quasi-Suslin weak duals

AbstractCascales, Ka̧kol, and Saxon (CKS) ushered Kaplansky and Valdivia into the grand setting of Cascales/Orihuela spaces E by proving:(K)If E is countably tight, then so is the weak space (E,σ(E,E′)), and(V)(E,σ(E,E′)) is countably tight iff weak dual (E′,σ(E′,E)) is K-analytic. The ensuing flow of quasi-Suslin weak duals that are not K-analytic, a la Valdivia's example, continues here, where we argue that locally convex spaces E with quasi-Suslin weak duals are (K, V)'s best setting: largest by far, optimal vis-a-vis Valdivia. The vaunted CKS setting proves not larger, in fact, than Kaplansky's. We refine and exploit the quasi-LB strong dual interplay

Some topological cardinal inequalities for spaces Cp(X)

Using the index of Nagami we get new topological cardinal inequalities for spaces Cp(X). A particular case of Theorem 1 states that if L ⊆ Cp(X) is a Lindelöf Σ-space and the Nagami index Nag(X) of X is less or equal than the density d(L) of L (which holds for instance if X is a Lindelöf Σ-space), then (i) there exists a completely regular Hausdorff space Y such that Nag(Y ) Nag(X), L ⊂ Cp(Y ) and d(L) = d(Y ); (ii) Y admits a weaker completely regular Hausdorff topology τ such that w(Y , τ ) d(Y ) = d(L). This applies, among other things, to characterize analytic sets for the weak topology of any locally convex space E in a large class G of locally convex spaces that includes (DF)-spaces and (LF)-spaces. The latter yields a result of Cascales–Orihuela about weak metrizability of weakly compact sets in spaces from the class G.The research was supported for the second named author by National Center of Science, Poland, Grant No. N N201 605340 and for the third author by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.Ferrando, JC.; Kakol, J.; López Pellicer, M.; Muñoz, M. (2013). Some topological cardinal inequalities for spaces Cp(X). Topology and its Applications. 160(10):1102-1107. https://doi.org/10.1016/j.topol.2013.04.024S110211071601

The uniform bounded deciding property and the separable quotient problem

[EN] Saxon-Wilansky's paper "The equivalence of some Banach space problems" contains six properties equivalent to the existence of an infinite dimensional separable quotient in a Banach space with nice simplified proofs. In the frame of uniform bounded deciding property, we prove that for an infinite dimensional Banach space E the following properties are equivalents: 1) The unit sphere of E contains a dense and non uniform bounded deciding subset. 2) The unit sphere S of E contains a dense and non strong norming subset. 3) E admits an infinite dimensional separable quotient.López Alfonso, S.; Moll López, SE. (2019). The uniform bounded deciding property and the separable quotient problem. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(2):1223-1230. https://doi.org/10.1007/s13398-018-0543-7S122312301132Argyros, S.A., Dodos, P., Kanellopoulos, V.: Unconditional families in Banach spaces. Math. Ann. 341, 15–38 (2008)Fernández, J., Hui, S., Shapiro, H.: Unimodular functions and uniform boundedness. Publ. Mat. 33, 139–146 (1989)Font, V.P.: On exposed and smooth points of convex bodies in Banach spaces. Bull. London Math. Soc. 28, 51–58 (1996)Ka̧kol, J., López-Pellicer, M.: On Valdivia strong version of Nikodym boundedness property. J. Math. Anal. Appl. 446, 1–17 (2017)López-Alfonso, S., Mas, J., Moll, S.: Nikodym boundedness property and webs in $$\sigma$$ σ -algebras. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 110, 711–722 (2016)López-Alfonso, S.: On Schachermayer and Valdivia results in algebras of Jordan measurable sets. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 110, 799–808 (2016)Nygaard, O.: A strong uniform boundedness principle in Banach spaces. Proc. Am. Math. Soc. 129, 861–863 (2001)Schachermayer, W.: On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Dissertationes Math. (Rozprawy Mat.) 214, 33 (1982)Śliwa, W.: The separable quotient problem and the strongly normal sequences. J. Math. Soc. Jpn. 64, 387–397 (2012)Saxon, S.A., Wilansky, A.: The equivalence of some Banach space problems. Colloq. Math. 37, 217–226 (1977)Valdivia, M.: On certain barrelled normed spaces. Ann. Inst. Fourier 29, 39–56 (1979)Valdivia, M.: On Nikodym boundedness property. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 107, 355–372 (2013

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. Appl. 35, 239–251 (1990). doi: 10.1016/0166-8641(90)90109-FBierstedt K.D., Bonet J.: Stefan Heinrich’s density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces. Math. Nachr. 35, 149–180 (1988)Cascales B.: On K-analytic locally convex spaces. Arch. Math. 49, 232–244 (1987)Cascales B., Ka̧kol J., Saxon S.A.: Weight of precompact subsets and tightness. J. Math. Anal. Appl. 269, 500–518 (2002). doi: 10.1016/S0022-247X(02)00032-XCascales B., Ka̧kol J., Saxon S.A.: Metrizability vs. Fréchet–Urysohn property. Proc. Am. Math. Soc. 131, 3623–3631 (2003)Cascales B., Namioka I., Orihuela J.: The Lindelöf property in Banach spaces. Stud. Math. 154, 165–192 (2003). doi: 10.4064/sm154-2-4Cascales B., Oncina L.: Compactoid filters and USCO maps. J. Math. Anal. Appl. 282, 826–843 (2003). doi: 10.1016/S0022-247X(03)00280-4Cascales B., Orihuela J.: On compactness in locally convex spaces, Math. Z. 195(3), 365–381 (1987). doi: 10.1007/BF01161762Cascales B., Orihuela J.: On pointwise and weak compactness in spaces of continuous functions. Bull. Soc. Math. Belg. Ser. B 40(2), 331–352 (1988) Journal continued as Bull. Belg. Math. Soc. Simon StevinDiestel J.: $${L^{1}_{X}}$$ is weakly compactly generated if X is. Proc. Am. Math. Soc. 48(2), 508–510 (1975). doi: 10.2307/2040292van Douwen E.K.: Prime mappings, number of factors and binary operations. Dissertationes Math. (Rozprawy Mat.) 199, 35 (1981)Drewnowski L.: Resolutions of topological linear spaces and continuity of linear maps. J. Math. Anal. Appl. 335(2), 1177–1195 (2007). doi: 10.1016/j.jmaa.2007.02.032Engelking R.: General Topology. Heldermann Verlag, Lemgo (1989)Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. Canadian Mathematical Society. Springer, Berlin (2001)Ferrando J.C.: A weakly analytic space which is not K-analytic. Bull. Aust. Math. Soc. 79(1), 31–35 (2009). doi: 10.1017/S0004972708000968Ferrando J.C.: Some characterization for υ X to be Lindelöf Σ or K-analytic in term of C p (X). Topol. Appl. 156(4), 823–830 (2009). doi: 10.1016/j.topol.2008.10.016Ferrando J.C., Ka̧kol J.: A note on spaces C p (X) K-analytic-framed in $${\mathbb{R}^{X} }$$ . Bull. Aust. Math. Soc. 78, 141–146 (2008)Ferrando J.C., Ka̧kol J., López-Pellicer M.: Bounded tightness conditions and spaces C(X). J. Math. Anal. Appl. 297, 518–526 (2004)Ferrando J.C., Ka̧kol J., López-Pellicer M.: A characterization of trans-separable spaces. Bull. Belg. Math. Soc. Simon Stevin 14, 493–498 (2007)Ferrando, J.C., Ka̧kol, J., López-Pellicer, M.: Metrizability of precompact sets: an elementary proof. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat. RACSAM 99(2), 135–142 (2005). http://www.rac.es/ficheros/doc/00173.pdfFerrando J.C., Ka̧kol J., López-Pellicer M., Saxon S.A.: Tightness and distinguished Fréchet spaces. J. Math. Anal. 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. Appl. 141, 171–186 (2004). doi: 10.1016/j.topol.2003.12.005Valdivia M.: Topics in Locally Convex Spaces, Mathematics Studies 67. North-Holland, Amsterdam (1982)Valdivia M.: Espacios de Fréchet de generación débilmente compacta. Collect. Math. 38, 17–25 (1987)Valdivia M.: Resolutions of identity in certain Banach spaces. Collect. Math. 38, 124–140 (1988)Valdivia M.: Resolutions of identity in certain metrizable locally convex spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.) 83, 75–96 (1989)Valdivia M.: Projective resolutions of identity in C(K) spaces. Arch. Math. (Basel) 54, 493–498 (1990)Valdivia, M.: Resoluciones proyectivas del operador identidad y bases de Markusevich en ciertos espacios de Banach. Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.) 84, 23–34Valdivia M.: Quasi-LB-spaces. J. Lond. Math. Soc. 35(2), 149–168 (1987). doi: 10.1112/jlms/s2-35.1.149Walker, R.C.: The Stone-Čech compactification Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 83. Springer, Berlin (1974

On topological properties of Fréchet locally convex spaces

[EN] We describe the topology of any cosmic space and any N-o-space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology sigma(E, E') are N-o-spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) l(1) and satisfying the Heinrich density condition we prove that (E, sigma(E,E')) is an N-o-space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain l(1), then (E, sigma(E, E')) is an N-o-space if and only if E' is separable. The last part of the paper studies the question: Which spaces (E, sigma(E, E')) are N-o-spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF)-space whose strong dual E' is separable, then (E, sigma(E, E')) is an N-o-space. Supplementing an old result of Corson we show that, for a Cech-complete Lindelof space X the following are equivalent: (a) X is Polish, (b) C-c(X) is cosmic in the weak topology, (c) the weak*-dual of C-c(X) is an N-o-space.The second and fourth named authors were supported by Generalitat Valenciana, Conselleria d'Educacio, Cultura i Esport, Spain, Grant PROMETEO/2013/058.Gabriyelyan, S.; Kakol, JM.; Kubzdela, A.; López Pellicer, M. (2015). On topological properties of Fréchet locally convex spaces. Topology and its Applications. 192(1):123-137. https://doi.org/10.1016/j.topol.2015.05.075S123137192

Weakly K-analytic spaces and the three-space property for analyticity

AbstractLet (E,E′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E′) to stronger ones in the frame of (E,E′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included

Note about lindelof Sigma-SPACES nu X

The paper deals with the following problem: characterize Tichonov spaces X whose realcompactification ¿X is a Lindelöf ¿-space. There are many situations (both in topology and functional analysis) where Lindelöf ¿ (even K-analytic) spaces ¿X appear. For example, if E is a locally convex space in the class fraktur G sign in sense of Cascales and Orihuela (fraktur G sign includes among others (LM)-spaces and (DF)-spaces), then ¿(E¿,¿(E¿,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales-Kakol-Saxon): if E¿fraktur G sign, then ¿(E¿,E) is K-analytic if and only if ¿(E¿,E) is Lindelöf. We prove a corresponding result for spaces Cp(X) of continuous real-valued maps on X endowed with the pointwise topology: ¿X is a Lindelöf ¿-space if and only if X is strongly web-bounding if and only if Cp(X) is web-bounded. Hence the weak* dual of C p(X) is a Lindelöf ¿-space if and only if Cp(X) is web-bounded and has countable tightness. Applications are provided. For example, every E¿fraktur G sign is covered by a family {A¿ :¿¿¿} of bounded sets for some nonempty set ¿¿¿¿. © Copyright Australian Mathematical Publishing Association Inc. 2011.This research is supported by the project of Ministry of Science and Higher Education, Poland, grant no. N 201 2740 33 and project MTM2008-01502 of the Spanish Ministry of Science and Innovation.Kakol, J.; López Pellicer, M. (2012). Note about lindelof Sigma-SPACES nu X. Bulletin of the Australian Mathematical Society. 85(1):114-120. https://doi.org/10.1017/S000497271100270XS114120851FERRANDO, J. C., & KĄKOL, J. (2008). A NOTE ON SPACES Cp(X)K-ANALYTIC-FRAMED IN ℝX. Bulletin of the Australian Mathematical Society, 78(1), 141-146. doi:10.1017/s0004972708000567Orihuela, J. (1987). Pointwise Compactness in Spaces of Continuous Functions. Journal of the London Mathematical Society, s2-36(1), 143-152. doi:10.1112/jlms/s2-36.1.143Okunev, O. G. (1993). On Lindelöf Σ-spaces of continuous functions in the pointwise topology. Topology and its Applications, 49(2), 149-166. doi:10.1016/0166-8641(93)90041-bCascales, B., & Orihuela, J. (1987). On compactness in locally convex spaces. Mathematische Zeitschrift, 195(3), 365-381. doi:10.1007/bf01161762Arkhangel’skii, A. V. (1992). Topological Function Spaces. Mathematics and Its Applications. doi:10.1007/978-94-011-2598-7Cascales, B. (1987). OnK-analytic locally convex spaces. Archiv der Mathematik, 49(3), 232-244. doi:10.1007/bf01271663Nagami, K. (1969). Σ-spaces. Fundamenta Mathematicae, 65(2), 169-192. doi:10.4064/fm-65-2-169-192Cascales, B., Ka̧kol, J., & Saxon, S. A. (2002). Weight of precompact subsets and tightness. Journal of Mathematical Analysis and Applications, 269(2), 500-518. doi:10.1016/s0022-247x(02)00032-xFerrando, J. C. (2009). Some characterizations for υX to be Lindelöf Σ or K-analytic in terms of Cp(X). Topology and its Applications, 156(4), 823-830. doi:10.1016/j.topol.2008.10.016Ferrando, J. C., Ka̧kol, J., López Pellicer, M., & Saxon, S. A. (2008). Quasi-Suslin weak duals. Journal of Mathematical Analysis and Applications, 339(2), 1253-1263. doi:10.1016/j.jmaa.2007.07.081Cascales, B., Kąkol, J., & Saxon, S. A. (2003). Proceedings of the American Mathematical Society, 131(11), 3623-3632. doi:10.1090/s0002-9939-03-06944-2Ka̧kol, J., & López Pellicer, M. (2007). Compact coverings for Baire locally convex spaces. Journal of Mathematical Analysis and Applications, 332(2), 965-974. doi:10.1016/j.jmaa.2006.10.045Valdivia, M. (1987). Quasi-LB-Spaces. Journal of the London Mathematical Society, s2-35(1), 149-168. doi:10.1112/jlms/s2-35.1.14