Probabilistic normed spaces

This book provides a comprehensive foundation in Probabilistic Normed (PN) spaces for anyone conducting research in this field of mathematics and statistics. It is the first to fully discuss the developments and the open problems of this highly relevant topic, introduced by A.N. Serstnev in the early 1960s as a response to problems of best approximations in statistics. The theory was revived by Claudi Alsina, Bert Schweizer and Abe Sklar in 1993, who provided a new, wider definition of a PN space which quickly became the standard adopted by all researchers. This book is the first wholly up-to-date and thorough investigation of the properties, uses and applications of PN spaces, based on the standard definition. Topics covered include: What are PN spaces? The topology of PN spaces. Probabilistic norms and convergence. Products and quotients of PN spaces. D-boundedness and D-compactness. Normability. Invariant and semi-invariant PN spaces. Linear operators. Stability of some functional equations in PN spaces. Menger's 2-probabilistic normed spaces. The theory of PN spaces is relevant as a generalization of deterministic results of linear normed spaces and also in the study of random operator equations. This introduction will therefore have broad relevance across mathematical and statistical research, especially those working in probabilistic functional analysis and probabilistic geometry.

1. Preliminaries. 1.1. Probability spaces. 1.2. Distribution functions. 1.3. The space of distance of distribution functions. 1.4. Copulas. 1.5. Triangular norms. 1.6. Triangle functions. 1.7. Multiplications. 1.8. Probabilistic metric spaces. 1.9. L[symbol] and Orlicz spaces. 1.10. Domination. 1.11. Duality -- 2. Probabilistic Normed spaces. 2.1. Probabilistic Normed spaces. 2.2. 1993: PN spaces redefined. 2.3. Special classes of PN spaces. 2.4. [symbol]-simple spaces. 2.5. EN spaces. 2.6. Probabilistic inner product spaces. 2.7. Open questions -- 3. The topology of PN spaces. 3.1. The topology of a PN space. 3.2. The uniform continuity of the probabilistic norm. 3.3. A PN space as a topological vector space. 3.4. Completion of PN spaces. 3.5. Probabilistic metrization of generalized topologies. 3.6. TIGT induced by probabilistic norms -- 4. Probabilistic norms and convergence. 4.1. The L[symbol] and Orlicz norms. 4.2. Convergence of random variables -- 5. Products and quotients of PN spaces. 5.1. Finite products. 5.2. Countable products of PN spaces. 5.3. Final considerations. 5.4. Quotients -- 6. D-boundedness and D-compactness. 6.1. The probabilistic radius. 6.2. Boundedness i…