The purpose of this book is to make available to beginning graduate students, and to others, some core areas of analysis which serve as prerequisites for new developments in pure and applied areas. We begin with a presentation (Chapters 1 and 2) of a selection of topics from the theory of operators in Hilbert space, algebras of operators, and their corresponding spectral theory. This is a systematic presentation of interrelated topics from infinite-dimensional and non-commutative analysis; again, with view to applications. Chapter 3 covers a study of representations of the canonical commutation relations (CCRs); with emphasis on the requirements of infinite-dimensional calculus of variations, often referred to as Ito and Malliavin calculus, Chapters 4–6. This further connects to key areas in quantum physics. Key Features: o Our book entails a varied choice of interdisciplinary themes. While the topics involved can be found in various parts of the pure and applied literature, there is a need for an accessible presentation which cuts across fields. In particular, we have aimed at "translating" between lingo which tends to be popular in different areas; keeping track of lingo used in diverse areas o As a result, our target audience is diverse as well, both with regards to level, and choice of topics. On the traditional mathematics side, we aim to reach users, at multiple levels, and with diverse interests, students and others who want to learn about exciting connections to applied problems in analysis, stochastic processes, dynamical system, representation theory, and mathematical physics

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