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Full Description
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.
Contents
Chapter 1 Ap'eritif: a Bit of Pre-functional Analysis.- Chapter 2 Norms and Inner Products.- Chapter 3 Completeness and Banach spaces.- Chapter 4 Bounded Linear Maps.- Chapter 5 Hahn-Banach Theorems.- Chapter 6 Hilbert Spaces and Orthogonality.- Chapter 7 Linear Functionals on a Hilbert Space
.- Chapter 8 Principle of Uniform Boundedness.- Chapter 9 The Polish quartet: open mapping and siblings
.- Chapter 10 The Polish Quartet.- Chapter 11 Finite dimensional spaces.- Chapter 12 Hilbert Space Operators.- Chapter 13 Dual spaces.- Chapter 14 Function spaces 1: Continuous functions.- Chapter 15 Compact operators 1 - Basics.- Chapter 16 Compact operators 2 - Spectral theorem.- Chapter 17 Banach algebras and the spectrum.- Chapter 18 Linear Functionals on C0(X).- Chapter 19 Spectral theorem for bounded operators.- Chapter 20 Convexity.- Chapter 21 Weak topologies.- Chapter 22 Extreme points and the Krein-Milman theorem.
