iv CONTENTS

Chapter 8. Integrability 43

8.1. Integrability of Φ 43

8.2. Integrability of Φ b Φ 44

8.3. Integrability of the inner product 48

Chapter 9. A Parseval-like formula for xx, yy, x P

lp,

y P

lq

51

Chapter 10. Grothendieck-like theorems in dimensions ą 2? 57

10.1. A multidimensional version of Question 2.3 57

10.2. A multidimensional version of Question 2.6 58

Chapter 11. Fractional Cartesian products and multilinear functionals on a

Hilbert space 61

11.1. Projective boundedness and projective continuity 61

11.2. Fractional Cartesian products 64

11.3. A characterization of projectively continuous functionals 68

Chapter 12. Proof of Theorem 11.11 73

12.1. A multilinear Parseval-like formula 73

12.2. The left-side inequality in (11.81) 82

12.3. The right-side inequality in (11.81) 84

12.4. Multilinear extensions of the Grothendieck inequality 85

Chapter 13. Some loose ends 87

13.1.

˜

V 2pX ˆ Y q vs.

˜

V 2pX ˆ Y q, G2pX ˆ Y q vs. G2pX ˆ Y q 87

13.2. G2pX ˆ Y q “

˜

V 2pX ˆ Y q vs. G2pX ˆ Y q “

˜

V 2pX ˆ Y q 87

13.3. Projective boundedness vs. projective continuity 87

13.4. A characterization of projective boundedness 88

Bibliography 89