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The closed graph theorem and the space of Henstock-Kurzweil integrable functions with the Alexiewicz norm. (English) Zbl 1267.54018

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.

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
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