## Christopher E. Heil

School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160
(404) 894-9231
h e i l @ m a t h . g a t e c h . e d u

C. Heil:   METRICS, NORMS, INNER PRODUCTS, AND OPERATOR THEORY, Birkhäuser/Springer, Cham, 2018 (xxi + 359 pp.).

Mathematics deals with many types of objects---shapes, numbers, points in Euclidean space, functions, linear transformations, graphs, to name only a few. However, these objects do not exist in isolation, and by looking at the properties of a related family or space of objects we often gain new understanding. In analysis, which in many ways is the "science of functions", the three main types of families that we encounter are metric spaces, normed spaces, and inner product spaces. This text is a self-contained introduction to these spaces, and to the operators that transform objects in one space (which are often themselves functions) into objects in another space. The intended reader of this text is a motivated student who is ready to take an upper-level, proof-based undergraduate mathematics course. No knowledge of measure theory or advanced real analysis is required.

• Preface.

• See the book at:
• Book reviews:
• MathSciNet. Quote: "The text is clearly and carefully written and is enhanced by an extensive list of problems, presented in increasing order of difficulty."

• Zentralblatt Math. Quote: "Each subchapter is endowed with a number of exercises. These vary in difficulty and type. Very interesting, for the classroom as well as for independent study, is the combination of more computational problems ... proof-oriented problems ... and problems that help students to think ..."

• Text Resources:

• A Solutions Manual for Instructors is available for instructors who register at the Birkhäuser website.

• The current errata list (updated July, 2021). Please email me with any typos or errors that you may find!

• Extra online Chapter 8: Integral Operators. While the main text does not require knowledge of Lebesgue measure and the Lebesgue integral, this chapter includes results that do depend on that material.

• A collection of Extra Material and Problems is available. This is a somewhat random assembly of results, problems, extra sections, and other material that is related to the text but did not make it into the text proper.

• A very limited "Cliff Notes" version of just some of the material from Chapters 1-5 of the text is available in the following manuscript:

• Handout: A Short Guide to Writing Proofs for beginning proof-writers (5 pages, 2011)

• Handout: A Short Review of Cardinality (6 pages, 2017)

• Handout: A brief review of Lebesgue Measure and the Lebesgue Integral (13 pages, 2017)

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