1 Introduction ................................................. 5
1.1 Basic notation .......................................... 7
1.2 Linear and Banach spaces ................................ 7
1.3 Summing norms and summing operators ..................... 9
1.4 Tensor norms ........................................... 12
2 Basic facts on multi-normed spaces .......................... 14
2.1 Multi-normed spaces .................................... 14
2.2 Multi-norms as tensor norms ............................ 15
2.3 The (p, q)-multi-norm .................................. 17
2.4 The (p, p)-multi-norm .................................. 19
2.5 Relations between (p, g)-multi-norms ................... 19
2.6 The standard t-multi-norm on Lr-spaces ................. 24
2.7 The Hilbert multi-norm ................................. 25
2.8 Relations between multi-norms .......................... 26
3 Comparing (p, g)-multi-norms on Lr spaces ................... 27
3.1 The case where r = 1 ................................... 27
3.2 The case where r > 1 ................................... 28
3.3 The role of Orlicz's theorem ........................... 30
3.4 Asymptotic estimates ................................... 30
3.5 Classification theorem ................................. 31
3.6 The role of Khintchine's inequalities .................. 33
3.7 Final classification ................................... 35
3.8 The relation with standard t-multi-norms ............... 38
4 The Hilbert space multi-norm ................................ 39
4.1 Equivalent norms ....................................... 39
4.2 Equivalence at level n ................................. 40
4.3 Calculation of c2 ...................................... 43
4.4 Calculation of c3 ...................................... 44
4.5 Calculation of c4 ...................................... 51
References ..................................................... 52
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