This core textbook on functional analysis is intended for senior undergraduates and graduate mathematics students. It is suitable for both classrooms and for self-study. The first in a two-volume series presents all the basic material needed for a solid foundation in the subject. It opens with a concise overview of the historical evolution of the subject and goes on to present foundational material in a clear, succinct manner, integrating original source quotes to enrich the narrative and blending the historical perspectives harmoniously with the flow of the subject. Pedagogically, the short chapters are more conducive for learning, and each chapter concludes with applications (including some unusual ones) to diverse fields and exercises to hone students’ understanding. The applications can also serve as sources for student seminars. Various formulations of the spectral theorem and their equivalence are discussed. Different approaches to some important results are presented to enrich the toolkit of the students. The style is neither terse nor verbose, requiring occasional paper-pencil work from the reader. The bibliography is rich and includes all the original works of the founding fathers. Thumbnail biographies of the mathematicians involved should pep up the readers.

Krishnan Parthasarathy is former Director and Head of the Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India. He earned his doctoral degree from the Indian Institute of Technology Kanpur and done his schooling and college education in Chennai (earlier Madras). His areas of research are abstract harmonic analysis and the theory of frames. He had taught subjects ranging from algebraic number theory to algebraic topology, differential equations to differential geometry, linear algebra to Lie algebras for about 35 years at the postgraduate level at different institutions. He had been a doctoral adviser for several students and has published a number of research papers in international journals of repute. He is a reviewer for several journals as well as on Mathematical Reviews and zbMATH.