Méndez, Luis Ángel Gutiérrez; Reyna, Juan Alberto Escamilla; Cárdenas, Maria Guadalupe Raggi; García, Juan Francisco Estrada The closed graph theorem and the space of Henstock-Kurzweil integrable functions with the Alexiewicz norm. (English) Zbl 1267.54018 Abstr. Appl. Anal. 2013, Article ID 476287, 4 p. (2013). Authors’ abstract: We prove that the cardinality of the space \({\mathcal HK}([a,b])\) is equal to the cardinality of the real numbers. Based on this fact we show that there exists a norm on \({\mathcal HK}([a,b])\) under which it is a Banach space. Therefore if we equip \({\mathcal HK}([a,b])\) with the Alexiewicz topology then \({\mathcal HK}([a,b])\) is not K-Suslin, neither infra-(u) nor a webbed space. Reviewer: Ryszard Pawlak (Łódź) Cited in 1 Document MSC: 54C35 Function spaces in general topology 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 46E15 Banach spaces of continuous, differentiable or analytic functions Keywords:Banach space; Henstoch-Kurzweil integrable function; graph of function; webbed (De Wilde) space; barrelled space; Suslin space; cardinality PDFBibTeX XMLCite \textit{L. Á. G. Méndez} et al., Abstr. Appl. Anal. 2013, Article ID 476287, 4 p. (2013; Zbl 1267.54018) Full Text: DOI References: [1] Swartz, C., Introduction to Gauge Integrals, x+157 (2001), Singapore: World Scientific, Singapore · Zbl 0982.26006 · doi:10.1142/9789812810656 [2] Merino, J. L. G., Integraciones de Denjoy de funciones con valores en espacios de Banach [Ph.D. thesis] (1997), Universidad Complutense de Madrid [3] Höning, C. S., There is no natural Banach space norm on the space of Kurzweil-Henstock-Denjoy-Perron integrable functions, Proceedings of the 30th Seminario Brasileiro de Análise [4] Pérez Carreras, P.; Bonet, J., Barrelled Locally Convex Spaces. Barrelled Locally Convex Spaces, North-Holland Mathematics Studies, 131, xvi+512 (1987), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0614.46001 [5] Köthe, G., Topological Vector Spaces. II. Topological Vector Spaces. II, Grundlehren der Mathematischen Wissenschaften, 237, xii+331 (1979), New York, NY, USA: Springer, New York, NY, USA · Zbl 0417.46001 [6] Ferrando, J. C., On convex-Suslin spaces, Acta Mathematica Hungarica, 55, 3-4, 201-206 (1990) · Zbl 0757.46007 · doi:10.1007/BF01950929 [7] Kruse, A. H., Badly incomplete normed linear spaces, Mathematische Zeitschrift, 83, 314-320 (1964) · Zbl 0117.08201 · doi:10.1007/BF01111164 [8] Nakamura, M., On quasi-Souslin space and closed graph theorem, Proceedings of the Japan Academy, 46, 514-517 (1970) · Zbl 0216.40604 · doi:10.3792/pja/1195520268 [9] Zhu, Q. D.; Zhao, Z. X., A-complete spaces and the closed graph theorem, Acta Mathematica Sinica, 24, 6, 833-836 (1981) · Zbl 0509.46005 [10] Pap, E.; Swartz, C., On the closed graph theorems, Generalized Functions and Convergence, 355-359 (1990), Teaneck, NJ, USA: World Scientific, Teaneck, NJ, USA [11] Beattie, R.; Butzmann, H.-P., Countability, completeness and the closed graph theorem, Generalized Functions, Convergence Structures, and Their Applications, 375-381 (1988), New York, NY, USA: Plenum Press, New York, NY, USA · Zbl 0744.46003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.