Contents
Preface vii
Chapter 1. Basic s o n 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
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
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 infinit e 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 typ e a 4 0
1.4.1. Genera l propertie s o f a-form s 4 2
1.4.2. Genera l propertie s o f o-integra l operator s 4 7
v
1.4.3. Compositio n o f a-integra l 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 o f 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
2.3. Injectiv e an d projectiv e (g)-norm s 8 4
2.4. Formatio n o f ne w (g)-norm s 9 0
2.5. Complement s o n /A , A\, / A \, \V , V/, \ V / 10
2.6. A tabl e o f natura l 0-norm s 106
Chapter 3 . (g)-norm s relate d t o Hilber t spac e 1
3.1. Definition s an d generalitie s abou t H and H * 1
3.2. Hermitia n H-form s 19
3.3. Hermitia n H*-form s 122
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