Preface ......................................................... v
1 Preliminaries ................................................ 1
1.1 Probability Spaces ...................................... 1
1.2 Distribution Functions .................................. 2
1.3 The Space of Distance of Distribution Functions ......... 4
1.4 Copulas ................................................. 6
1.5 Triangular Norms ........................................ 7
1.6 Triangle Functions ...................................... 8
1.7 Multiplications ........................................ 11
1.7.1 The multiplication τт ........................... 11
1.7.2 The multiplication Пт ........................... 12
1.7.3 Convolution ..................................... 12
1.7.4 Convolution-related operations and random
variables ....................................... 12
1.8 Probabilistic Metric Spaces ............................ 13
1.9 V and Orlicz Spaces .................................... 14
1.10 Domination ............................................. 17
1.11 Duality ................................................ 20
2 Probabilistic Normed Spaces ................................. 23
2.1 The First Definition ................................... 23
2.2 1993: PN Spaces Redefined .............................. 24
2.3 Special Classes of PN Spaces ........................... 29
2.3.1 Equilateral spaces .............................. 29
2.3.2 Simple PN spaces ................................ 30
2.4 a-simple Spaces ........................................ 30
2.5 EN Spaces .............................................. 38
2.6 Probabilistic Inner Product Spaces ..................... 40
2.7 Open Questions ......................................... 45
3 The Topology of PN Spaces ................................... 47
3.1 The Topology of a PN Space ............................. 47
3.2 The Uniform Continuity of the Probabilistic Norm ....... 48
3.3 A PN Space as a Topological Vector Space ............... 49
3.4 Completion of PN Spaces ................................ 54
3.5 Probabilistic Metrization of Generalized Topologies .... 61
3.6 TIGT Induced by Probabilistic Norms .................... 62
4 Probabilistic Norms and Convergence ......................... 67
4.1 The Lp and Orlicz Norms ................................ 67
4.2 Convergence of Random Variables ........................ 68
5 Products and Quotients of PN Spaces ......................... 75
5.1 Finite Products ........................................ 75
5.2 Countable Products of PN Spaces ........................ 76
5.2.1 The σ-product ................................... 76
5.2.2 The τ-product ................................... 80
5.3 Final Considerations ................................... 81
5.4 Quotients .............................................. 83
5.4.1 Completeness results ............................ 85
6 P-Boundedness and  -Compactness ............................. 91
6.1 The Probabilistic Radius ............................... 91
6.2 Boundedness in PN Spaces ............................... 93
6.3 Total Boundedness ..................................... 112
6.4 D-Compact Sets in PN Spaces ........................... 117
6.5 Finite Dimensional PN Spaces .......................... 118
7 Normability ................................................ 123
7.1 Normability of Šerstnev Spaces ........................ 123
7.2 Other Cases ........................................... 125
7.3 Normability of PN Spaces .............................. 129
7.4 Open Questions ........................................ 139
8 Invariant and Semi-Invariant PN Spaces ..................... 141
8.1 Invariance and Semi-Invariance ........................ 141
8.2 New Class of PN Spaces ................................ 145
8.3 Open Questions ........................................ 149
9 Linear Operators ........................................... 151
9.1 Boundedness of Linear Operators ....................... 151
9.2 Classes of Linear Operators ........................... 160
9.3 Probabilistic Norms for Linear Operators .............. 161
9.4 Completeness Results .................................. 167
9.5 Families of Linear Operators .......................... 169
10 Stability of Some Functional Equations in PN Spaces ........ 173
10.1 Mouchtari-Šerstnev Theorem ............................ 173
10.2 Stability of a Functional Equation in PN Spaces ....... 175
10.3 The Additive Cauchy Functional Equation in RN Spaces:
Stability ............................................. 181
10.4 Stability in the Quartic Functional Equation
in RN Spaces .......................................... 187
10.4.1 The quartic functional equation in RN
spaces: stability .............................. 187
10.5 A Functional Equation in Menger PN Spaces ............. 191
10.5.1 Probabilistic stability of the functional
equation (10.5.1) .............................. 192
11 Menger's 2-Probabilistic Normed Spaces ..................... 197
11.1 Accretive Operators in 2-PN Spaces .................... 200
11.2 Convex Sets in 2-PN Spaces ............................ 203
11.3 Compactness and Boundedness in 2-PN Spaces ............ 205
11.4 P-Boundedness in 2-PN Spaces .......................... 209
Bibliography .................................................. 211
Index ......................................................... 219
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