Contents Preface vii Chapter 1 . Basic s on tenso r norm s 1 The algebrai c preliminarie s 1 1.1. Reasonabl e crossnorms , includin g th e norm s A and V 5 1.1.1. Definition s 5 1.1.2. Injectivit y o f V and projectivit y o f A 1 1 A A 1.1.3. Th e universa l mappin g propert y o f S an d th e dua l o f X g Y 1 3 1.1.4. Examples : C(K) g X an d L X (M) ® x 1 4 v 1.1.5. Integra l bilinea r form s an d th e dua l o f X g Y 2 2 1.2. Definitio n o f (g)-norm s 2 5 1.2.1. Fundamenta l operation s o n (g)-norm s 2 6 1.2.2. Orde r relation s amon g (g-norm s 2 8 1.3. Extensio n o f (g-norm s t o space s o f infinite dimension s 2 9 1.3.1. Metri c accessibilit y an d accessibilit y 3 2 1.4. Bilinea r form s an d linea r operator s o f type a 4 0 1.4.1. Genera l propertie s o f a-forms 4 2 1.4.2. Genera l propertie s o f o-integral operator s 4 7 v 1.4.3. Compositio n o f a-integral an d a-integra l operator s 4 8 1.4.4. Accessibilit y an d metri c accessibilit y (continued ) 5 0 1.5. a-nuclea r form s an d operator s 5 4 1.6. Th e Dvoretzky-Roger s theorem , Grothendieck-styl e 5 9 1.6.1. Th e fundamenta l lemm a 5 9 1.6.2. Consequence s 6 3 Chapter 2 . Th e rol e of C(K)-spaces an d L 1 -spaces 6 7 2.1. Complement s o n A and V 6 7 2.1.1. Representability , equimeasurabilit y an d nuclearit y 7 2 2.2. Fundamenta l linea r topologica l propertie s o f C- an d L-space s 7 6 Notes 8 1 2.3. Injectiv e an d projectiv e (g)-norm s 8 4 2.4. Formatio n o f new (g)-norm s 9 0 2.5. Complement s o n /A , A\, / A \, \V , V/, \ V / 10 1 2.6. A table o f natural 0-norm s 10 6 Chapter 3 . (g)-norm s related t o Hilber t spac e 11 1 3.1. Definition s an d generalitie s abou t H and H * 11 1 3.2. Hermitia n H-form s 11 9 3.3. Hermitia n H*-form s 12 2
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