The reviewer praises Barbara MacCluer's 'Elementary Functional Analysis' as a concise yet comprehensive introduction to the subject, suitable for beginning graduate students. They highlight the book's methodical proofs, well-chosen examples, and extensive exercises, particularly in operator theory and spectral analysis, making it a strong contender for canonical status in the field.
The reviewer commends Barbara MacCluer's 'Elementary Functional Analysis' for its concise yet thorough coverage of the subject, aimed at beginning graduate students. The book starts with basic Banach and Hilbert space theory, emphasizing geometric properties, and includes a non-trivial example of a Hilbert space using complex analysis. The reviewer appreciates the methodical proofs and well-chosen examples, particularly in operator theory and spectral analysis, which are the book's strongest points. The extensive exercises, totaling 192, are noted for their appropriate difficulty and relevance, though the reviewer wishes for more significant applications of the ideas presented. Overall, the book is praised for its accessibility and the author's personal touch, including anecdotes about the mathematicians who developed the theory, making it a valuable resource for self-learners.
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.