Functional Analysis and the Feynman Operator Calculus 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 Feynmans 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.