Topological vector spaces, h h schaefer, m p wolff

Graduate Texts in Mathematics Editorial Board S Axler F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 TAKEUTilZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nded HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHESIPIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules 2nd ed GoLUBITSKY/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nded HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNEs/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEwITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEy General Topology ZARISKI/SAMUEL Commutative Algebra Vol.I ZARISKI/SAMUEL Commutative Algebra Vol.II JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nded 35 ALExANoERIWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 APoSTOL Modular Functions and Dirichlet Series in Number Theory 2nded 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEVE Probability Theory I 4th ed 46 LoEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHslWu General Relativity for Mathematicians 49 GRUENBERo/WEIR Linear Geometry 2nded 50 EowAROS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELLlFox Introduction to Knot Theory 58 KOBUTZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory (continued after index) H.H Schaefer With M.P Wolff Topological Vector Spaces Second Edition Springer H.H Schaefer M.P Wolff Eberhard-Karls-Universitat Tiibingen Mathematisches Institut Auf der Morgenstelle 10 Tiibingen, 72076 Germany Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA K A Ribet Mathematics Department Universityof California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (1991): 46-01, 46Axx, 46Lxx Library of Congress Cataloging-in-Publication Data Schaefer, Helmut H Topological vector spaces - 2nd ed / Helmut H Schaefer in assistance with M Wolff p cm - (Graduate texts in mathematics ; 3) Includes bibliographical references and indexes ISBN 978-1-4612-7155-0 ISBN 978-1-4612-1468-7 (eBook) DOI 10.1007/978-1-4612-1468-7 Linear topological spaces Wolff, Manfred, 1939II Title III Series QA322.S28 1999 Sl4'.32-dc21 98-53842 Printed on acid-free paper First edition © 1966 by H H Schaefer Published by the Macmillan Company, New York © 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1999 Softcover reprint ofthe hardcover 2nd edition 1999 AlI rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media New York, except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage' and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especialIy identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Alian Abrams; manufacturing supervised by Jeffrey Taub 54 32 ISBN 978-1-4612-7155-0 PREFACE TO THE SECOND EDITION As the first edition of this book has been well received through five printings over a period of more than thirty years, we have decided to leave the material of the first edition essentially unchanged - barring a few necessary updates On the other hand, it appeared worthwhile to extend the existing text by adding a reasonably informative introduction to C* - and W* -algebras The theory of these algebras seems to be of increasing importance in mathematics and theoretical physics, while being intimately related to topological vector spaces and their orderings-the prime concern of this text The authors wish to thank J Schweizer for a careful reading of Chapter VI, and the publisher for their care and assistance Tiibingen, Germany Spring 1999 H H Schaefer M P Wolff v Preface The present book is intended to be a systematic text on topological vector spaces and presupposes familiarity with the elements of general topology and linear algebra The author has found it unnecessary to rederive these results, since they are equally basic for many other areas of mathematics, and every beginning graduate student is likely to have made their acquaintance Similarly, the elementary facts on Hilbert and Banach spaces are widely known and are not discussed in detail in this book, which is :plainly addressed to those readers who have attained and wish to get beyond the introductory level The book has its origin in courses given by the author at Washington State University, the University of Michigan, and the University of Ttibingen in the years 1958-1963 At that time there existed no reasonably ccmplete text on topological vector spaces in English, and there seemed to be a genuine need for a book on this subject This situation changed in 1963 with the appearance of the book by Kelley, Namioka et al [1] which, through its many elegant proofs, has had some influence on the final draft of this manuscript Yet the two books appear to be sufficiently different in spirit and subject matter to justify the publication of this manuscript; in particular, the present book includes a discussion of topological tensor products, nuclear spaces, ordered topological vector spaces, and an appendix on positive operators The author is also glad to acknowledge the strong influence of Bourbaki, whose monograph [7], [8] was (before the publication of Kothe [5]) the only modern treatment of topological vector spaces in printed form A few words should be said about the organization of the book There is a preliminary chapter called "Prerequisites," which is a survey aimed at clarifying the terminology to be used and at recalling basic definitions and facts to the reader's mind Each of the five following chapters, as well as the Appendix, is divided into sections In each section, propositions are marked u.v, where u is the section number, v the proposition number within the vi PREFACE vii section Propositions of special importance are additionally marked "Theorem." Cross references within the chapter are (u.v), outside the chapter (r, u.v), where r (roman numeral) is the number of the chapter referred to Each chapter is preceded by an introduction and followed by exercises These "Exercises" (a total of 142) are devoted to further results and supplements, in particular, to examples and counter-examples They are not meant to be worked out one after the other, but every reader should take notice of them because of their informative value We have refrained from marking some of them as difficult, because the difficulty of a given problem is a highly subjective matter However, hints have been given where it seemed appropriate, and occasional references indicate literature that may be needed, or at least helpful The bibliography, far from being complete, contains (with few exceptions) only those items that are referred to in the text I wish to thank A Pietsch for reading the entire manuscript, and A L Peressini and B J Walsh for reading parts of it My special thanks are extended to H Lotz for a close examination of the entire manuscript, and for many valuable discussions Finally, I am indebted to H Lotz and A L Peressini for reading the proofs, and to the publisher for their care and cooperation H.H.S Tiibingen, Germany December, 1964 Table if Contents Preface to the Second Edition Preface v vi Prerequisites A Sets and Order B General Topology C Linear Algebra I TOPOLOGICAL VECTOR SPACES Introduction Vector Space Topologies 12 12 Product Spaces Subspaces Direct Sums Quotient Spaces 19 Topological Vector Spaces of Finite Dimension 21 Linear Manifolds and Hyperplanes 24 Bounded Sets 25 Metrizability 28 Complexification 31 33 Exercises II LOCALL Y CONVEX TOPOLOGICAL VECTOR SPACES Introduction 36 Convex Sets and Semi-Norms 37 Normed and Normable Spaces 40 The Hahn-Banach Theorem 45 viii ix TABLE OF CONTENTS Locally Convex Spaces Projective Topologies Inductive Topologies Barreled Spaces Bornological Spaces Separation of Convex Sets 10 Compact Convex Sets Exercises III LINEAR MAPPINGS Introduction Continuous Linear Maps and Topological Homomorphisms Banach's Homomorphism Theorem Spaces of Linear Mappings Equicontinuity The Principle of Uniform Boundedness and the Banach-Steinhaus Theorem Bilinear Mappings Topological Tensor Products Nuclear Mappings and Spaces Examples of Nuclear Spaces 47 51 54 60 61 63 66 68 The Approximation Property Compact Maps Exercises 73 74 76 79 82 87 92 97 106 108 115 IV DU ALITY Introduction Dual Systems and Weak Topologies Elementary Properties of Adjoint Maps Locally Convex Topologies Consistent with a Given Duality.The Mackey-Arens Theorem Duality of Projective and Inductive Topologies Strong Dual of a Locally Convex Space Bidual 122 123 128 130 133 Reflexive Spaces 140 Dual Characterization of Completeness Metrizable Spaces Theorems of Grothendieck, Banach-Dieudonne, and Krein-Smulian 147 BIBLIOGRAPHY 335 (See also Grosberg, J and M Krein.) [1] Sur quelques questions de la geometrie des ensembles convexes situes dans un espace lineaire norme et complet C R (Doklady) Acad Sci URSS (N.S.), 14 (1937), 5-8 [2] Proprietes fondamentales des ensembles coniques normaux dans l'espace de Banach C R (Doklady) Acad Sci URSS (N.S.), 28 (1940), 13 -17 KREIN, M KREIN, M AND D MILMAN [1] On extreme points of regular convex sets Studia Math., (1940), 133-138 KREIN, M G AND M A RUTMAN [1] Linear operators leaving invariant a cone in a Banach space Uspehi Mat Nauk (N.S.) 3, no (23) (1948), 3-95 (Russian) Also Amer Math Soc Transl no 26 (1950) KREIN, M AND V §MULIAN [1] On regularly convex sets in the space conjugate to a Banach space Ann of Math (2) 41 (1940), 556-583 LANDSBERG, M [1] Pseudonormen in der Theorie der linearen topologischen Raume Math Nachr., 14 (1955), 29-38 [2] Lineare topologische Raume, die nicht lokalkonvex sind Math Z., 65 (1956), 104-112 MACKEY, G W [1] On infinite dimensional linear spaces Proc Nat Acad Sci U.S.A., 29 (1943), 216-221 [2] On convex topological linear spaces Proc Nat Acad Sci U.S.A., 29 (1943), 315-319 [3] Equivalence of a problem in measure theory to a problem in the theory of vector lattices Bull Amer Math Soc., 50 (1944), 719-722 [4] On infinite-dimensional linear spaces Trans Amer Math Soc., 57 (1945), 155-207 [5] On convex topological linear spaces Trans Amer Math Soc., 60 (1946), 519-537 MAC LANE, s (See Birkhoff, G., and S Mac Lane.) MAHOWALD, M [1] Barrelled spaces and the closed graph theorem J London Math Soc., 36 (1961), 108-110 MARTINEAU, A [1] Sur une propriete caracteristique d'un produit de droites Arch Math., 11 (1960), 423-426 MAURIN, K [1] Abbildungen vom Hilbert-Schmidtschen Typus und ihre Anwendungen Math Scand., (1961), 359-371 (See also Krein, M., and D P Milman.) [1] Characteristics of extreme points of regularly convex sets Dokl Akad Nauk SSSR (N.S.), 57 (1947), 119-122 MILMAN, D P 336 BIBLIOGRAPHY NACHBIN, L [1] Topological vector spaces of continuous functions Proc Nat A cad Sci U.S.A., 40 (1954), 471-474 (See also Kelley, J L., I Namioka, and co-authors.) [1] Partially ordered linear topological spaces Mem Amer Math Soc no 24 (1957) NAMIOKA, I [2] A substitute for Lebesgue's bounded convergence theorem Proc Amer Math Soc 12 (1961), 713-716 V NEUMANN, J [1] On complete topological linear spaces Trans Amer Math Soc., 37 (1935), 1-20 PERESSINI, A L [1] On topologies in ordered vector spaces Math Ann., 144 (1961), 199-223 [2] Concerning the order structure of Kothe sequence spaces Michigan Math J 10 (1963), 409-415 [3] A note on abstract (M)-spaces Illinois J Math., (1963), 118-120 PERRON, O [1] Zur Theorie der Matrices Math Ann., 64 (1907), 248-263 PHILLIPS, R S (See Hille, E and R S Phillips.) PIETSCH, A [1] Unbedingte und absolute Summierbarkeit in F-Riiumen Math Nachr., 23 (1961), 215-222 [2] Verallgemeinerte vollkommene Folgenriiume Berlin 1962 [3] Zur Theorie der topologischen Tensorprodukte Math Nachr 25 (1963), 19-30 [4] Eine neue Charakterisierung der nuklearen lokalkonvexen Riiume I Math Nachr., 25 (1963), 31-36 [5] Eine neue Charakterisierung der nuklearen lokalkonvexen Riiume II Math Nachr., 25 (1963), 49-58 [6] Absolut summierende Abbildungen in lokalkonvexen Riiumen Math Nachr., 27 (1963), 77-103 [7] Zur Fredholmschen Theorie in lokalkonvexen Riiumen Studia Math., 22 (1963), 161-179 [8] Nukleare lokalkonvexe Riiume Berlin 1965 POULSEN, E T [1] Convex sets with dense extreme points Amer Math Monthly, 66 (1959), 577-578 PTAK, v [1] On complete topological linear spaces Czechoslovak Math J.,3 (78) (1953), 301-364 [2] Compact subsets of convex topological linear spaces Czechoslovak Math J., (79) (1954),51-74 [3] Weak compactness in convex topological linear spaces Czechoslovak Math J.,4 (79) (1954), 175-186 [4] On a theorem of W F Eberlein Studia Math., 14 (1954), 276-284 337 BIBLIOGRAPHY [6] Two remarks on weak compactness Czechoslovak Math J., (80) (1955), 532-545 [6] Completeness and the open mapping theorem Bull Soc Math France, 86 (1958),41-74 [7] A combinatorial lemma on the existence of convex means and its application to weak compactness Proc Sympos Pure Math., Vol VI/ Convexity, 437-450 Providence, R I 1963 RIESZ, F [I] Sur quelques notions fondamentales dans la theorie generale des operations Iineaires Ann of Math., (2) 41 (1941), 174-206 ROBERTSON, A AND W ROBERTSON [1] On the closed graph theorem Proc Glasgow Math Assoc., (1956), 9-12 [2] Topological vector spaces Cambridge 1964 ROBERTSON, W [1] Contributions to the general theory of linear topological spaces Thesis, Cambridge 1954 ROGERS, C A (See Dvoretzky, A and C A Rogers.) ROTA, G.-C [1] On the eigenvalues of positive operators Bull Amer Math Soc., 67 (1961), 556-558 RUTMAN, M A (See Krein, M G and M A Rutman.) SCHAEFER, H H [1] Positive Transformationen in lokalkonvexen halbgeordneten Vektorraumen Math Ann., 129 (1955), 323-329 [2] Halbgeordnete lokalkonvexe Vektorraume Math Ann., 135 (1958), 115-141 [3] Halbgeordnete lokalkonvexe Vektorraume II Math Ann., 138 (1959), 259-286 [4] Halbgeordnete lokalkonvexe Vektorraume III Math Ann., 141 (1960), 113-142 [5] On the completeness of topological vector lattices Michigan Math J., (1960), 303-309 [6] Some spectral properties of positive linear operators Pacific J Math., 10 (1960), 1009-1019 [7] On the singularities of an analytic function with values in a Banach space Arch Math 11 (1960), 40-43 [8] Spectral measures in locally convex algebras Acta Math., 107 (1962), 125173 [9] Convex cones and spectral theory Proc Sympos Pure Math., Vol VII Convexity, 451-47l Providence, R I 1963 [10] Spektraleigenschaften positiver Operatoren Math Z., 82 (1963),303-313 [11] On the point spectrum of positive operators Proc Amer Math Soc., 15 (1964), 56-60 SCHAEFER, H H AND B J WALSH [1] Spectral operators in spaces of distributions Bull Amer Math Soc., 68 (1962), 509-511 338 BIBLIOGRAPHY SCHATIEN, R [1] A theory of cross spaces Ann Math Studies no 26 (1960) [2] Norm ideals of completely continuous operators Berlin-Gottingen-Heidelberg 1960 SCHAUDER, J [1] Zur Theorie stetiger Abbildungen in Funktionalriiumen Math Z., 26 (1927), 47-65 und 417-431 [2] Ober lineare stetige Funktionaloperationen Studia Math., (1930), 183-196 SCHWARTZ, J T (See Dunford, N and J T Schwartz.) SCHWARTZ, L (See also Dieudonne, J et L Schwartz.) [1] Theorie des distributions Tome I 2nd ed Paris 1957 [2] Theorie des distributions Tome II 2nd ed Paris 1959 SHIROTA, T [1] On locally convex vector spaces of continuous functions Proc Jap Acad., 30 (1954) 294-298 ~MULIAN, v L (See also Krein, M and V.Smulian.) [1] Sur les ensembles faiblement compacts dans les espaces lineaires normes Comm Inst Sci Mat Mec Univ Charkov (4), 14 (1937), 239-242 [2] Sur les ensembles regulierement fermes et faiblement compacts dans les espaces du type (B) C R (Doklady) Acad Sci UR$S (N.S.) 18, (1938), 405-407 [3] Ober lineare topologische Riiume Mat Sbornik N.S., (49) (1940),425-448 STEINHAUS, H (See Banach, S et H Steinhaus.) STONE, M H [1] The generalized Weierstrass approximation theorem Math Mag 21 (1948), 167-183 and 237-254 SZ.-NAGY, B [1] Spektraldarstellung linearer Transformationen des Hilbertschen Riiumes Berlin-Gottingen-Heidelberg 1942 TAYLOR, A E [1] Introduction to functional analysis New York 1958 TYCHONOFF, A [1] Ein Fixpunktsatz Math Ann., 111 (1935), 767-776 WALSH, B J (See Schaefer, H H and B J Walsh.) WEHAUSEN, V [1] Transformations in linear topological spaces Duke Math J., (1938), 157-169 (See Civin, P and B Yood.) YOOD, B ADDITIONAL BIBLIOGRAPHY BRATTELI, O AND D W ROBINSON [l] Operator algebras and quantum statistical mechanics, Vols I and II New York 1979 and 1981 CALKIN, J W [l] Two-sided ideals and congruences in the ring of bounded operators in Hilbert space Ann of Math (2) 42 (1941), 839-873 CONNES, A [1] Une classification des facteurs de type III Ann Sci Ecole Norm Sup (4) (1973), 133-252 [2] Sur la classification des facteurs de type II C.R Acad Sci Paris Ser A-B 281 (1975), A13-AI5 CUNTZ, J [1] Simple C*-algebras generated by isometries Comm Math Phys 57 (1977), 173-185 DIXMIER, J [1] Von Neumann algebras Amsterdam 1981 [2] C* -algebras Amsterdam 1982 ENFLO, P [1] A counterexample to the approximation problem in Banach spaces Acta Math 130 (1973), 309-317 GELFAND, I [l] Normierte Ringe Mat Sb (1941), 1-24 GELFAND, I AND M A NAIMARK [1] On the embedding of normed rings into the ring of operators in Hilbert space Mat Sb 12 (1943), 197-213 IONESCU TULCEA, A AND C [1] Topics in the theory of liftings Berlin Heidelberg New York 1969 339 340 ADDITIONAL BIBLIOGRAPHY KADISON, R V AND J R RINGROSE [1] Fundamentals of the theory of operator algebras, Vols I and II New York 1983 KAPLANSKY, I [1] A theorem on rings of operators Pacific J Math (1951),227-232 LINDENSTRAUSS, J AND L TZAFRIRI [1] Classical Banach spaces, I Berlin Heidelberg New York 1977 [2] Classical Banach spaces, II Berlin Heidelberg New York 1979 MURPHY, G M [1] C* -algebras and operator theory Boston 1990 MURRAY, F J AND J VON NEUMANN [1] On rings of operators Ann of Math 37 (1936), 116-229 [2] On rings of operators, II Trans Amer Math Soc 41 (1937),208-248 [3] On rings of operators, IV Ann of Math (2) 44 (1943), 716-808 VON NEUMANN, J [1] Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren Math Ann 102 (1929), 370-427 [2] On rings of operators, III Ann of Math 41 (1940),94-161 PEDERSEN, G [1] CO-algebras and their automorphism groups New York 1979 POWERS, R T [1] Representations of uniformly hyperfinite algebras and their associated von Neumann rings Ann of Math (2) 86 (1967), 138-171 SAKAI, s [l] CO-algebras and W*-algebras Berlin Heidelberg New York 1971 SCHAEFER, H H [12] Banach lattices and positive operators Berlin Heidelberg New York 1974 SEGAL, I [1] Irreducible Representations of Operator Algebras Bull A.M.S 53 (1947), 7388 TAKESAKI, M [1] Theory of operator algebras, I New York 1979 INDEX NOTE: This index is not intended to assist the reader in surveying the subject matter of the book (for this, see table of contents), but merely to help him locate definitions and look up theorems that have acquired a special name The latter are collectively listed under "theorem"; other compound expressions are, in principle, listed by their qualifying attributes For example, "topology of compact convergence" would be found under "compact convergence" rather than "topology." Amplification, 287 (AM)-space, 242 with unit, 242 Analytic vector-valued function, 200-1 Approximate unit, 269 Approximation problem, 108 Approximation property (= a.p.) Archimedean order, 205 Associated bijective map, 10 Associated bomological space, 63 Associated Hausdorff t.v.s., 20 Associated weak topology, 52 Automorphism, 262 inner, 263 Absolute, 207 Absolute value, 10 Absolute polar, 125 Absolutely convergent series, 120 Absolutely summable family, 120 Absorb, 11 Absorbing, II Addition, Adherent point of filter, of set, Adjoint (linear map), 111, 128, 155 algebraic, 128 Affine subspace, 17 (AL)-space, 242 Algebra Banach, 259 C*-,260 Calkin, 300 Cunlz,302 factor algebra, 293 locally convex, 202 non-degenerate, 288 normed,259 ordered, 255 reduced group, 261 spectral, 255 Toeplilz, 301 unital, 259 von Neumann, 288 W*-, 261, 277 Algebraic adjoint, 128 Algebraic dual, 10 Almost Archimedean order, 254 Almost uniform convergence, 121 B*-algebra,261 Baire space, Banach algebra, 202 Banach algebra, 259 Banach lattice, 235 Banach space, 41 Band, 209 Barrel, 60 Barreled space, 60 Basis (Hamel basis), 10, 21 Basis problem, 115 B-complete, 162 B,-complete, 162 Bi-bounded convergence (topology), 173 Bidual, 143 strong, 143 Bi-eqnicontinuous convergence (topology), 91, 96 Bijection, Bijective, 341 342 Bilinear form, 88 Bilinear map, 87 Bipolar, 126 Biunivocal, Bomological space, 61 Bound greatest lower, least upper, lower, upper, Bounded convergence (topology), 81 Bounded filter, 86 Bounded map, 98 Bounded set, 25 (B)-space, 41 C' -algebra, 261 Calkin algebra, 300 Canonical bilinear form, 123 Canonical bilinear map, 92 Canonical decomposition, 10 Canonical imbedding, 10, 55, 97, 143, 182 Canonical isomorphism, 167, 183 Canonical map, 97, 172 (quotient), Canonical ordering, 206 Category first, second, Cauchy filter, Cauchy sequence, C-compact linear map, 314 Center, 292 Centrally orthogonal, 296 Circled, 11 Circled hull, 11, 39 Closed map, Closed set, Closed, convex, circled hull, 39 Closed, convex hull, 39 Closure, Cluster point, Coarser filter, Coarser ordering, Coarser topology, Coarser unifonnity, Codimension, 22 Commutant, 288 Compact convergence (topology), 80 Compact map, 98 Compact set, Complement, Complementary subspace, 20 Complete lattice, Complete uniform space, Completely regular, Completion of topological vector lattice, 235 of t.v.s., 17 of uniform space, Complex extension, 261 Complexification, 33 Condensation of singularities, 117 Cone, 38,215 of compact base, 72 INDEX Conjugate-linear, 45 Conjugate map, 45 Conjugation-invariant, 31, 245 Connected, Consistent (locally convex topology), 130 Contact point of filter, sequence, directed family, of set, Continuous, Convergent filter, sequence directed family, 4-5 Convex body, 40 Convex, circled, compact convergence (topology), 81 Convex, circled hull, 39 Convex cone, 38 Convex function, 68 Convex hull, 39 Convex set, 37 Countably compact, 185 Cross-norm, 119 C-saturated,215 C-spectral radius, 266 Cuntz algebra, 302 Cyclic peripheral point spectrum, 271 Dense, (DF)-space, 154 Dimension, 10,21 Direct image (topology), Direct sum algebraic, 19 oft.v.s., 19, 33 Directed family, of semi-norms, 69 Directed set, Disconnected, Discrete topology, Disjoint (lattice), 207 Distingnished I.c.s., 193 Domain, Dual algebraic, 10, 24 strong (ofl.c.s.), 141-46 oft.v.s.,48 weak, 52 Dual cone, 218 Duality, 123 Dual system, 123 Eigenspace, 308 Eigenvalue, 308 Eigenvector, 308 Elementary filter, 4, 117 Entourage, Equicontinuous, 82 separately, 88 Equivalent projections, 293-4 Evaluation map, 143 Extension, Extremally disconnected, 280 Extreme boundary, 275 Extremally disconnected, 247 Extreme point, 67 Extreme ray, 72 343 INDEX Factor, 293 Family, Filter, Filter base, Finer filter, Finer ordering, Finer topology, Finer uniformity, Finest locally convex topology, 56 Finite rank (linear map), 98 (F)-lattice, 235 Frechet lattice, 235 Frechet space, 49 Fredholm operator, 301 (F)-space, 49 Function strongly continuous, 291 Fundamental family, 25, 79 Fundamental system, 25 Gauge, 39 Generated (subspace), 10 Generated (topology), 48 Generating cone, 205 Generating family of semi-norms, 48 Gestufter Raum, 120 Graph, I Greatest lower bound, Hamel basis, 10, 21 Hardy space, 30 I Hausdorff space, Hermitian, 261 Hermitian form, 44, 273 Hilbert dimension, 44 Hilbert direct sum, 45 Hilbert space, 44-45 Holomorphic, locally, 201 Homeomorphic, Homeomorphism, Homomorphism, 262 Homomorphism (topological), 75 Hull circled, 39 convex, 39 C-saturated, 217 saturated,81 Hull topology, Hyperplane, 24 real, 32 Hypocontinuous, 89 Ideal,260 left,260 proper, 260 right, 260 two-sided, 260 Induced order, 206 Induced topology, Induced uniformity, Inductive limit, 57 Inductive topology, 5, 54 of tensor product, 96 Inductively ordered, Infimum, Infinite tensorproduct, 305 Infrabarreled, 142 Injective, Injection, 55 Inner product space, 44 Integral bilinear form, 169 Integral linear map, 169 Interior (set), Interior point, Inverse image (topology), Involution, 260 Irreducible positive endomorphism, 317 Isometry partial, 293 Isomorphic (t.v.s.), 13 Isomorphic (uniform space), Isomorphic (vector space), 10 (ordered vector space), 205 Isomorphism, 262 W'-isomorphism, 286 Jordan decomposition, 272 Kernel, 10 Kernel topology, Lattice, Lattice disjoint, 207 Lattice homomorphism, 213 Lattice isomorphism, 213 Lattice ordered, 214 Lattice semi-norm, 235 (LB)-space, 58 L.c.s (= locally convex space), 47 L.c.v.l (= locally convex vector lattice), 235 Least upper bound, Left vector space, Lexicographical order, 210 (LF)-space, 58 Limit, Line segment, 37 Linear combination, Linear form, 10 real, 32 Linear form normal,285 order continuous, 285 Linear hull, 10 Linear manifold, 24 Linear map, 10 Linearly independent, 10 Locally bounded, 30 Locally compact, Locally convex algebra, 202 Locally convex direct sum, 55 Locally convex space, 47 Locally convex topology, 47 Locally convex vector lattice, 235 Locally holomorphic, 201 Locally solid, 234 Lower bound, Mackey space, 132 Mackey topology, 131 Majorant,3 344 Majorized, Maximal,3 Meager, Metric, 7-8 Metric space, Metrizable topological space, Metrizable t.v.s., 28 Minimall.c.s., 191 Minimal topology, 132, 191 Minimal type (vector lattice), 213 Minkowski functional, 39 Minorant,3 Minorized, Modulus, 269 Monotone transfinite sequence, 253 Montel space, 147 Multilinear, 119 Multiplicity of eigenvalues, 308 Natural topology (bidual), 143 Nearly open map, 163 Neighborhood, Neighborhood base, Neighborhood filter, Norm, 39, 40 Norm isomorphic, 41 Norm isomorphism, 41 Normable space, 41 Normal cone, 215 Normal (element), 261 Normal topological space, Normal topology, 190 Normed lattice, 235 Normed space, 40 Non-discrete, 11 Non-meager, Nowhere dense, Nuclear linear map, 98 Nuclear space, 100 Null space, 10 Open set, Open map, Operational calculus, 304 Operator, 306 Operator Fredholm, 301 normal,303 Toeplitz, 301 Order, Order bidual, 212 Order bound dual, 205, 214 Order bounded, 3, 205 Order (C' -algebra), 268 canonical, 268 Order complete, 209 Order convergent, 238 Order dual, 206, 214 Order interval, 205 Order limit, 238 Order structure, Order summable, 231 Order topology, 230 Order unit, 205 Ordered algebra, 255 Ordered direct sum, 206 INDEX Ordered set, Ordered vector space over C, 214 over R, 204 topological, 222 Ordering, canonical, 206 Orthogonal, 44 Orthogonal projection, 44 Orthogonal subspace (duality), 127 Orthonormal basis, 44 Parallel,24 Partial isometry, 293 Perfect space, 190 Peripheral spectrum, 316 Point spectrum, 308 peripheral, 316 Polar, 125 absolute, 125 Polar decomposition, 293 Positive cone, 205 Positive definite Hermitian form, 44 Positive element, 205 Positive (element), 269 Positive face of dual unit ball, 247 Positive linear form, 206, 216 Positive (linear form), 270 Positive linear map, 225 Positive sequence of type 11, 231 Precompact, 8, 25 Precompact convergence (topology), 81 Pre-Hilbert space, 44 Prenuclear family, 178 Prenuclear semi-norm, 177 Prenuclear set, 177 Pre-order, 250 Principal part (Laurent expansion), 308 Product topology, Product (oft.v.s.), 19 Product uniformity, Projection, 19, 52 orthogonal, 44 Projection, 274 abelian, 297 central, 295 finite, 297 infinite, 297 purely infinite, 297 Projective limit, 52 Projective topology, 5, 51 of tensor product, 93 Proper cone, 205 Pseudo-norm, 28 Ptak space, 162 Quasi-complete, 27 Quotein map, Quotient set, Quotient space (t.v.s.), 20 Quotient topology, 5, 20 Radial,11 Radon measure, 43 Range, Rank, 92, 98 345 INDEX Rare, Reduced projective limit, 139 Reflexive, 144 Regular order, 206 Regular topological space, Representation, 273 cyclic, 273 faithful, 273 irreducible, 302 unitarily equivalent, 273 universal, 274 Residue, 260 Resolution of the identity, 275, 303 Resolvent, 259 set, 259 Resolvent equation, 259 Resolvent set, 259 Restriction, Saturated family, 81 Saturated hull (of family of sets), 81 Scalar multiplication, Schauder basis, 114 Schwarz' inequality, 44 -cone, 217 -convergence (topology), 79 Section of directed family, of ordered set, Section filter, Self-adjoint (element), 261 Self-adjoint (linear form), 270 Semi-complete, Semi-norm, 39 Semi-reflexive, 143 Semi-space, 63-64 Separable, Separated topological space, Separated uniformity, Separately continuous, 88 Separately equicontinuous, 88 Separating hyperplane, 64 Sequence, Sequentially complete, -hypocontinuous, 89 Simple convergence (topology), 81 Simply bounded, 82 Singleton (set), I Solid, 209 Spatially isomorphic, 273 Spectral algebra, 255 Spectral circle, 259 Spectral element, 256 Spectral mapping theorem, 260, 264 Spectral measure, 255 Spectral operator, 256 Spectral radius, 260 Spectrum, 260 peripheral point, 316 point, 308 State, 272 product, 305 space, 272 tracial, 304 Stonian space, 255 6-topology, 79 x :!-topology, 91 Strict inductive limit, 57 Strict 6-