The reviewer praises Barbara MacCluer's 'Elementary Functional Analysis' for its comprehensive coverage and accessible style, making it a strong contender for a canonical textbook in the field. They highlight the book's depth and the author's expertise, particularly in operator theory, as well as the well-chosen exercises that enhance understanding.
The reviewer highly recommends Barbara MacCluer's 'Elementary Functional Analysis' as a standout textbook in the field. They appreciate the book's concise yet thorough coverage of key topics, including Banach and Hilbert space theory, operator theory, and spectral theory. The reviewer notes that MacCluer's expertise in operator theory shines through, particularly in the chapters on compact operators and Banach algebras. The numerous well-chosen exercises are praised for their ability to deepen the reader's understanding and command of the subject. The reviewer also appreciates the personal touches, such as anecdotes about the mathematicians who developed the theory, which make the text more engaging. Overall, they believe the book successfully achieves its goal of providing a solid introduction to functional analysis and could gain recognition as a fine introduction to the subject.
Quick quotes
The topics covered are treated in full and the overall length of the text is kept under control through a careful selection of topics.
MacCluer’s discussion of ideals and homomorphisms of Banach algebras, commutative Banach algebras, weak topologies, the Banach-Alaoglu theorem, the Gelfand transform and the continuous functional calculus for normal operators is nothing short of excellent.
The exercises round out the material in the chapters nicely, involving the student enough to give him a sufficient command of the subject.