1 Introduction ................................................. 7
1.1 Basic notation .......................................... 7
1.1.1 Sets and sequences ............................... 7
1.1.2 Inequalities ..................................... 7
1.1.3 Linear spaces .................................... 8
1.1.4 Linear operators and matrices ................... 11
1.2 Banach spaces and Banach algebras ...................... 12
1.2.1 Banach spaces and operators ..................... 12
1.2.2 Tensor products ................................. 14
1.2.3 Direct sum decompositions ....................... 16
1.2.4 Duals of products of Banach spaces .............. 17
1.2.5 Families of Banach spaces ....................... 17
1.2.6 Hilbert spaces and C*-algebras .................. 18
1.2.7 Standard Banach spaces .......................... 19
1.2.8 Banach algebras ................................. 22
1.2.9 Hermitian elements .............................. 24
1.3 Banach lattices ........................................ 25
1.3.1 Definitions ..................................... 25
1.3.2 Complexifications ............................... 27
1.3.3 Continuity, boundedness and completeness ........ 30
1.3.4 Positive, regular, and order-bounded operators .. 31
1.3.5 The Banach algebra  r(E) ........................ 34
1.3.6 Dual Banach lattices ............................ 34
1.3.7 AL and AM spaces ................................ 36
1.4 Summary ................................................ 37
1.5 History and acknowledgements ........................... 38
2 The axioms and some consequences ............................ 40
2.1 The axioms ............................................. 40
2.1.1 Multi-norms ..................................... 40
2.1.2 Dual multi-norms ................................ 41
2.1.3 Independence of the axioms ...................... 41
2.2 Elementary consequences of the axioms .................. 43
2.2.1 Results for special-norms ....................... 43
2.2.2 Results for multi-norms ......................... 44
2.2.3 Results for dual multi-norms .................... 45
2.2.4 The family of multi-norms ....................... 46
2.2.5 Standard constructions .......................... 47
2.3 Theorems on duality .................................... 49
2.3.1 Special-normed spaces ........................... 49
2.3.2 Multi-normed and dual multi-normed spaces ....... 49
2.4 Reformulations of the axioms ........................... 51
2.4.1 Multi-norms and matrices ........................ 51
2.4.2 Dual multi-norms and matrices ................... 53
2.4.3 Generalizations ................................. 54
2.4.4 Sequential norms ................................ 55
2.4.5 Multi-norms and tensor norms .................... 55
3 The minimum and maximum multi-norms ......................... 58
3.1 An associated sequence ................................. 58
3.2 The minimum multi-norm ................................. 58
3.2.1 Definitions ..................................... 58
3.2.2 Finite-dimensional spaces ....................... 60
3.3 The maximum multi-norm ................................. 61
3.3.1 Existence of the maximum multi-norm ............. 61
3.3.2 The sequence (φnmax(E) .......................... 62
3.4 Summing norms .......................................... 63
3.4.1 Introduction .................................... 63
3.4.2 Summing constants ............................... 66
3.4.3 Related constants ............................... 68
3.4.4 Orlicz property ................................. 69
3.4.5 Specific spaces ................................. 70
3.5 Characterizations of the maximum multi-norm ............ 71
3.5.1 Characterizations in terms of weak summing
norms ........................................... 71
3.5.2 The dual of the minimum dual multi-norm ......... 73
3.5.3 Characterizations in terms of projective norms .. 74
3.6 The function φnmax for some examples ................... 76
3.6.1 The spaces ℓP ................................... 76
3.6.2 The spaces LP ................................... 78
3.6.3 The spaces C(K) ................................. 79
3.6.4 A lower bound for φnmax(E) ...................... 79
4 Specific examples of multi-norms ............................ 80
4.1 The (p, g)-multi-norm .................................. 80
4.1.1 Definition ...................................... 80
4.1.2 Relations between (p, g)-multi-norms ............ 82
4.1.3 Duality theory .................................. 84
4.1.4 The dual of the (p, g)-special-norm ............. 84
4.1.5 Multi-norms on Hilbert spaces ................... 86
4.2 Standard q-multi-norms ................................. 88
4.2.1 Definition ...................................... 88
4.2.2 A comparison of multi-norms ..................... 90
4.2.3 Maximality ...................................... 90
4.2.4 Equality of two multi-norms on L1(Ω) ........... 91
4.2.5 Equivalence of multi-norms on ℓP ................ 92
4.2.6 The spaces M(K) ................................. 94
4.2.7 The Schauder multi-norm ......................... 96
4.2.8 Abstract q-multi-norms .......................... 97
4.3 Lattice multi-norms .................................... 99
4.3.1 Multi-norms and Banach lattices ................. 99
4.3.2 A representation theorem ....................... 102
4.4 Summary ............................................... 103
5 Multi-topological linear spaces and multi-norms ............ 105
5.1 Basic sets ............................................ 105
5.1.1 Topological linear spaces ...................... 105
5.1.2 Multi-topological linear spaces ................ 105
5.2 Multi-null sequences .................................. 107
5.2.1 Convergence .................................... 107
5.2.2 Multi-normed spaces ............................ 108
5.2.3 Multi-null sequences and order-convergence ..... 111
6 Multi-bounded sets and multi-bounded operators ............. 113
6.1 Definitions and basic properties ...................... 113
6.1.1 Multi-bounded sets ............................. 113
6.1.2 Multi-bounded sets for lattice multi-norms ..... 114
6.1.3 Multi-bounded operators ........................ 115
6.1.4 Multi-continuous operators ..................... 117
6.2 The space  (E, F) ..................................... 117
6.2.1 The normed space (E, F) ....................... 117
6.2.2 A multi-norm based on (E, F) .................. 119
6.3 Examples .............................................. 121
6.3.1 Algebras of operators .......................... 121
6.3.2 Partition multi-norms .......................... 122
6.4 Multi-bounded operators on Banach lattices ............ 124
6.4.1 Multi-bounded and order-bounded operators ...... 124
6.4.2 The multi-bounded multi-norm ................... 129
6.5 Extensions of multi-norms ............................. 130
6.5.1 Definitions .................................... 130
6.5.2 Examples of balanced multi-normed spaces ....... 131
6.5.3 Examples of isometric multi-normed spaces ...... 131
7 Orthogonality and duality .................................. 133
7.1 Decompositions ........................................ 133
7.1.1 Hermitian decompositions of a normed space ..... 133
7.1.2 Small decompositions of multi-normed spaces .... 137
7.1.3 Orthogonal decompositions of multi-normed
spaces ......................................... 138
7.1.4 Elementary examples ............................ 141
7.1.5 Decompositions of the spaces C(K) .............. 142
7.1.6 Decompositions of Hilbert spaces ............... 143
7.1.7 Decompositions of lattices ..................... 143
7.1.8 Decompositions of Lp-spaces .................... 145
7.2 Multi-norms generated by closed families .............. 146
7.2.1 Generation of multi-norms ...................... 146
7.2.2 Orthogonality with respect to families ......... 148
7.2.3 Orthogonality and Banach lattices .............. 148
7.3 Multi-norms on dual spaces ............................ 150
7.3.1 The multi-dual space ........................... 150
7.3.2 Second dual spaces ............................. 152
References ................................................. 154
Index of terms ................................................ 158
Index of symbols .............................................. 163
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