Nonstandard order analysis. An invitation. (Q2703791)

This book presents the prerequisites for adaptation and application of the model-theoretic tools of nonstandard set theory to investigating \(K\)-spaces and classes of linear operators in them. NEWLINENEWLINENEWLINEChapter 1 collects information on the formal set theories most common in the contemporary research in functional analysis. The authors start with recalling the axiomatics of the classical Zermelo-Fraenkel set theory and von Neumann-Gödel-Bernays class theory. They then overview the Boolean valued models which stem from the works of D. Scott, R. Solovay, and P. Vopěnka. They also sketch Nelson's internal set theory and one of the most powerful and promising variants of external set theory which was recently proposed by T. Kawai and is widely used in modern infinitesimal analysis. The authors finish this chapter with brief exposition of relatively standard set theory due to E. I. Gordon and Y. Pŕaire. NEWLINENEWLINENEWLINEChapter 2 deals with Boolean valued analysis of vector lattices. It is well known by now that the most principal new nonstandard opportunity for \(K\)-spaces consists in formalizing the heuristic transfer principle by L. V. Kantorovich which claims that the members of every Dedekind complete vector lattice are generalized numbers. Boolean valued analysis demonstrates rigorously that the elements of every \(K\)-space depict reals in an appropriate nonstandard model of set theory. The formalism the authors set forth in this chapter belongs now to the list of the basic and compulsory conceptions of the theory of ordered vector spaces. NEWLINENEWLINENEWLINEChapter 3 treats infinitesimal constructions. A. Robinson's apology for the infinitesimal has opened new possibilities in Banach space theory from scratch. The central place is occupied by the concept of nonstandard hull of a normed space \(X\), i.e., the factor space of the external subspace of elements with limited norm by the monad set of members of \(X\) with infinitesimal norm. The authors discuss adaptation of nonstandard hulls to vector lattice theory in Section 3.1. They proceed to Section 3.2 with introducing another important construction of nonstandard analysis, the Loeb measure. NEWLINENEWLINENEWLINESections 3.3 and 3.4 address the still-uncharted topic of combing Boolean valued and infinitesimal methods. Two approaches seem theoretically feasible: the first may consist in studying a Boolean-valued model immersed into the inner universe of some external set theory. This approach is pursued in Section 3.3. The other approach consists in studying an appropriate fragment of some nonstandard set theory (for instance, in the form of ultraproduct or ultralimit) which lies within a relevant Boolean-valued universe. The authors take this approach in Section 3.4. It is worth of emphasizing that the corresponding formalisms, in spite of their affinity, lead to principally different constructions in \(K\)-space theory. The authors illustrate these particularities in technique by examining the `cyclic' topological notions of import for applied Boolean-valued analysis. NEWLINENEWLINENEWLINEChapter 4 deals with nonstandard analysis in operator theory. The authors first address positive linear operators which are listed among the central objects of the theory of ordered vector spaces. The main opportunity, offered by nonstandard methods, consists in the fact that the available formalism allows us to simplify essentially the analysis of operators and vector measures by reducing the environment to the case of functionals and scalar measures and sometimes even to ordinary numbers. NEWLINENEWLINENEWLINEIn Sections 4.1-4.4 the authors illustrate the general tricks of nonstandard analysis in operator theory in connection with the extension and decomposition problems as well as representing homomorphisms and Maharam operators. The authors also distinguish a new class of cyclically compact operators. The leave some place for the problem of generating the fragments of a positive operator. The matter is that the complete description rests on successive application of nonstandard analysis in the Boolean-valued and infinitesimal versions. The chapter is closed with Boolean-valued analysis of one of the most important facts of the classical theory of operator equations, the Fredholm alternative. The authors give its analog for a new class of equations with cyclicalllly compact operators. NEWLINENEWLINENEWLINEChapter 5 collects 88 unsolved problems from nonstandard analysis. NEWLINENEWLINENEWLINEThe book is written for graduate and postgraduate students studying nonstandard methods of analysis and for a general mathematical audience interested in applications of model-theoretic methods in functional analysis.