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Infinite dimensional analysis. A hitchhiker’s guide. (English) Zbl 0839.46001
Studies in Economic Theory. 4. Berlin: Springer-Verlag. xviii, 603 p. (1994).
The book is an excellent overview of the main results in infinite-dimensional analysis which may be applicable to economical problems. Infinite-dimensional models have become prominent in economics and finance because they capture natural aspects of the world that cannot be described and examined in finite-dimensional models. Dynamic models require infinite-dimensional function spaces. Another feature of the world that arguably requires infinite-dimensional modeling is uncertainty.
The approach used in the book is very modern and rigorous. The choice of the subjects is of course open to discussion, but from the reviewer’s point of view it has been made very successfully. The content can be estimated from the list of chapter titles: Odds and ends. Topology. Metrizable spaces. Topological vector spaces. Normed spaces. Riesz spaces. Banach lattices. Charges and Measures. Integrals. \(L_p\)-spaces. Measures and topology. Probability measures on metrizable spaces. Spaces of sequences. Correspondences. Markov transitions. Ergodicity.

46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
46A40 Ordered topological linear spaces, vector lattices
28A12 Contents, measures, outer measures, capacities
46B42 Banach lattices