Contents
Chapter 1. Introduction 1
1.1. The inequality 1
1.2. An integral representation 1
1.3. Parseval-like formulas 3
1.4. Multilinear Parseval-like formulas 4
1.5. Projective boundedness and projective continuity 7
1.6. A personal note and acknowledgements 8
Chapter 2. Integral representations: the case of discrete domains 9
2.1. First question 9
2.2. Second question 11
2.3. Third question and Grothendieck’s th´ eor` eme fondamental 14
Chapter 3. Integral representations: the case of topological domains 19
3.1.
L2-continuous
families 19
3.2. A “continuous”’ version of le th´ eor` eme fondamental 22
Chapter 4. Tools 25
4.1. The framework 25
4.2. Rademacher and Walsh characters 25
4.3. Walsh series 26
4.4. Riesz products 27
4.5. Continuity 29
Chapter 5. Proof of Theorem 3.5 31
5.1. The construction of Φ 31
5.2. Φ is odd 32
5.3. An integral representation of the dot product 33
5.4. Φ is weakly continuous 33
5.5. Φ is
pl2
Ñ
L2q-continuous
34
Chapter 6. Variations on a theme 35
6.1. The map Φ2 35
6.2. Sharper bounds 38
Chapter 7. More about Φ 39
7.1. Spectrum 39
7.2.
(l2
Ñ
Lp)-continuity
39
7.3.
(l2
Ñ
L8)-continuity?
40
7.4. Linearization 41
iii
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