THE OPEN MAPPING AND CLOSED GRAPH THEOREMS IN TOPOLOGICAL VECTOR SPACES

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1 THE OPEN MAPPING AND CLOSED GRAPH THEOREMS IN TOPOLOGICAL VECTOR SPACES BY TAQDIR HUSAIN MoMaster University SPRINGER FACHMEDIEN WIESBADEN GMBH 1965

2 Diese Arbeit wurde in Rahmen des Jubiläum8preisausschreibens des Verlages Friedr. Vieweg & Sohn mit einem Preis ausgezeichnet. Die Übersetzung in die englische Sprache besorgte der Verfasser This work won a prize in the Jubilee Prize Competition held by the Verlag Friedr. Vieweg & Sohn. The English translation has been made by the author ISBN ISBN (ebook) DOI / by Springer Fachmedien Wiesbaden Originally published by Friedr. VieweiJ & Sohn, Braunschweig in 1965

3 TO MARTHA

4 PREFACE THE main purpose of writing this monograph is to give a picture of the progress made in recent years in understanding three of the deepest results of Functional Analysis-namely, the open-mapping and closed theorem. graph theorems, and the so-called K r e i n - ~ m u l i a n In order to facilitate the reading of this book, some of the important notions and well-known results about topological and vector spaces have been collected in Chapter 1. The proofs of these results are omitted for the reason that they are easily available in any standard book on topology and vector spaces e.g. Bourbaki [2], Keiley [18], or Köthe [22]. The results of Chapter 2 are supposed to be weil known for a study of topological vector spaces as weil. Most of the definitions and notations of Chapter 2 are taken from Bourbaki's books [3] and [4] with some trimming and pruning here and there. Keeping the purpose of this book in mind, the presentation of the material is effected to give a quick resume of the results and the ideas very commonly used in this field, sacrificing the generality of some theorems for which one may consult other books, e.g. [3], [4], and [22]. From Chapter 3 onward, a detailed study of the open-mapping and closed-graph theorems as weil as the K r e i n - ~ theorem m u l ihas a been n carried out. For the arrangement of the contents of Chapters 3 to 7, see the Historical Notes (Chapter 8). The bibliography, which can be regarded by no means as complete, is given at the end of the Historical Notes. For detailed references one can consult [7], besides other books referred to above. Square brackets containing numerals indicate references in the bibliography. The reference of definitions of terms used in this book can be found in the index which is preceded by an index for notations and symbols. The author is greatly indebted to his wife, Martha Husain, for her devoted co-operation and assistance in preparing this book, especially in rendering the German translation of this book which, under the title On Topological V ector Spaces with Emphasis on the Open M apping and Olosed Graph Theorems, won a prize in the Jubilee Competition

5 Vlll PREFACE sponsored bythe Friedr. Vieweg & SohnPublishing House, Braunschweig, W. Germany, on its I 75th anniversary. The author takes great pleasure in thanking the Friedr. Vieweg & Sohn Publishing House for their award and kindness in agreeing to publish the book in co-operation with the Glarendon Press, Oxford. My thanks are also due to the editors and the staff of the Glarendon Press for producing this book so carefully. McMaster University Hamilton, Canada TAQDIR HUSAIN

6 CONTENTS 1. ELEMENTARY CONCEPTS CONCERNING TOPO- LOGICAL AND VECTOR SPACES 1. Definition of a topological space 2. Bases and sub-bases of a neighbourhood system 3. Metric spaces 4. Filters and compact sets 5. Uniform spaces and completeness 6. N owhere-dense sets and sets of first and second category 7. Topological products 8. Mappings 9. Vector spaces 2. TOPOLOGICAL VECTOR SPACES 1. Definition of a topological vector space 2. N eighbourhood systems in a TVS 3. The Hahn-Banachtheorem 4. Locally convex topological vector spaces 5. Inductive limits of l.c. spaces 6. Barrelied and bornological spaces 7. 6-topologies and the principle of uniform boundedness 8. The Banach-Steinhaus theorem 9. Duality theory l l THE OPEN-MAPPING AND CLOSED-GRAPH THEOREMS Finite-dimensional TVS's and the open-mapping theorem Banach's theorems Some generalizations of Theorems 3 and B-COMPLETENESS AND THE OPEN-MAPPING THEOREM Definition of B-complete spaces The v(u)-topology Ultra-barrelled spaces B,-complete l.c. spaces The open-mapping and closed-graph theorems for Br complete spaces THE ew*-topology AND VARIOUS NOTIONS OF COMPLETENESS 1. The k-extension of a topology in topological spaces 2. The k-extension of a topology in a topological vector space

7 X CONTENTS 3. The ew*-topology and completeness 4. The ew*-topology and pseudo-completeness 5. The cew*-topology and completeness 6. The ew*-topology and hypercompleteness 6. THE THEORY OF S-SPACES I. Definition and a characterization of S-spaces 2. S-spaces and B-completeness 3. The Krein-Smulian theorem 4. Subspaces and quotient spaces of an S-space 5. The dual of an S-space 6. Countability conditions 7. LOCALLY CONVEX SPACES WITH THE B(Cf/)- PROPERTY I. The concept of the B(Cf/)-property 2. The closed-graph theorem for B,.(Cf/ )-spaces 3. The Br(ß)- and the B(ß)-spaces 4. B(.Pß)-spaces and B(ß)-spaces 5. The closed-graph theorem for Br(ß)-spaces 6. Examples and counter-examples 8. HISTORICAL NOTES BIBLIOGRAPHY INDEX OF SYMBOLS INDEX

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