"The book is a self-contained treatise on the mathematical foundation of Feynman operational calculus and Feynman path integrals. ... It contains a large amount of original material which cannot be found elsewhere in book form; in fact, most of the original results are due to the authors themselves. ... The book will be of interest to both graduate students and researchers in pure or applied mathematics." (Sonia Mazzucchi, Mathematical Reviews, November, 2016)

"This book carries Fujiwara's insight to provide the mathematical frameworks for Feynman's operator calculus and for the Feynman path integral. ... The book is intended for advanced graduate students and researchers and can be used as a text for advanced courses in functional analysis, operator theory, mathematical physics, or related subjects." (Miyeon Kwon, zbMATH 1345.81001, 2016)>

Preliminary Background.- Integration on Infinite-dimensional Spaces.- HK-Integral and HK-Spaces.- Analysis on Hilbert Space.- Operators on Banach Space.- Spaces of von Neumann Type.- The Feynman Operator Calculus.- Applications of the Feynman Calculus.

This book provides the mathematical foundations for Feynman's operator calculus and for the Feynman path integral formulation of quantum mechanics as a natural extension of analysis and functional analysis to the infinite-dimensional setting. In one application, the results are used to prove the last two remaining conjectures of Freeman Dyson for quantum electrodynamics. In another application, the results are used to unify methods and weaken domain requirements for non-autonomous evolution equations. Other applications include a general theory of Lebesgue measure on Banach spaces with a Schauder basis and a new approach to the structure theory of operators on uniformly convex Banach spaces. This book is intended for advanced graduate students and researchers.

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