The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm

Luis Ángel Gutiérrez Méndez,Juan Alberto Escamilla Reyna,Maria Guadalupe Raggi Cárdenas,Juan Francisco Estrada García

doi:10.1155/2013/476287

Luis Ángel Gutiérrez Méndez, Juan Alberto Escamilla Reyna + Show 2 more

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https://doi.org/10.1155/2013/476287

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Abstract

We prove that the cardinality of the spaceℋ𝒦([a,b])is equal to the cardinality of real numbers. Based on this fact we show that there exists a norm onℋ𝒦([a,b])under which it is a Banach space. Therefore if we equipℋ𝒦([a,b])with the Alexiewicz topology thenℋ𝒦([a,b])is not K-Suslin, neither infra-(u) nor a webbed space.

Highlights

Let [a, b] be a compact interval in R
It is a known fact that if f : X → Y is a continuous function between topological spaces, where Y is a Hausdorff space, f has closed graph; the converse is not true in general
The versions of the Closed Graph Theorem establish under what conditions the converse is fulfilled

Summary

Introduction

Let [a, b] be a compact interval in R. It is a known fact that if f : X → Y is a continuous function between topological spaces, where Y is a Hausdorff space, f has closed graph; the converse is not true in general. The importance of these theorems is that in certain contexts it is easier to prove that a function has closed graph than to prove that it is continuous; Propositions 10, 12, and 14 are examples of this fact.

Results

Conclusion