Spaces of Continuous Functions
The space C(X) of all continuous functions on a compact space X carries the structure of a normed vector space, an algebra and a lattice. On the one hand we study the relations between these structures and the topology of X, on the other hand we discuss a
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G.L.M. Groenewegen A.C.M. van Rooij
Spaces of Continuous Functions
Atlantis Studies in Mathematics Volume 4
Series editor Jan van Mill, VU University, Amsterdam, The Netherlands
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G.L.M. Groenewegen A.C.M. van Rooij •
Spaces of Continuous Functions
G.L.M. Groenewegen Faculty of Education—ILS HAN University of Applied Sciences Nijmegen The Netherlands
ISSN 1875-7634 Atlantis Studies in Mathematics ISBN 978-94-6239-200-7 DOI 10.2991/978-94-6239-201-4
A.C.M. van Rooij Faculty of Science—Mathematics Radboud University Nijmegen The Netherlands
ISSN 2215-1885
(electronic)
ISBN 978-94-6239-201-4
(eBook)
Library of Congress Control Number: 2016936970 © Atlantis Press and the author(s) 2016 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
Preface This book is written in the spirit of Z. Semadeni’s Banach Spaces of Continuous Functions and Rings of Continuous Functions by L. Gillman and M. Jerison, but on a lower level. It is intended to be more accessible than Semadeni’s book – at the cost of being considerably less thorough. It covers a wider range than the Gillman-Jerison book, putting more stress on lattice structure, less on multiplication. Basically, our text deals with the connection between the topology of a space 𝑋 (mostly a compact Hausdorff space) and the algebraic structure of 𝐶(𝑋), the set of all continuous real valued functions on 𝑋. We consider 𝐶(𝑋) as a ring, as a lattice and as a metric space. A typical result is that 𝑋 (compact Hausdorff) is metrizable if and only if 𝐶(𝑋) is separable. Another: if 𝑋 and 𝑌 are compact Hausdorff spaces and if 𝐶(𝑋) and 𝐶(𝑌) are isomorphic rings, then 𝑋 and 𝑌 must be homeomorphic. A second theme is the occurrence of 𝐶(𝑋) (with compact 𝑋) in representation theorems. Not only is 𝐶(𝑋) an algebra (ring + vector space) and a Riesz space (lattice + vector space), several beautiful theorems characterize among the algebras and the Riesz spaces those that are “isomorphic” to 𝐶(𝑋) for some 𝑋. This leads to the Sto
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