Real and Functional Analysis

Going through Real and Functional Analysis, one has the impression that the book results from a very successfully completed graft of the branches of Rudin’s Functional Analysis (i.e. locally convex spaces and distributions, the Fourier transform and Sobolev spaces, Banach algebras, unbounded operators and operator semigroups) with the choicest fruits in the rich root network of Rudin’s Real and Complex Analysis (i.e. the fundamentals of measure theory, the Lebesgue integral, the fundamental theorem of calculus, a brief introduction of Hilbert spaces, and the “big” theorems of functional analysis).

 

The authors, Bogachev and Smolyanov, have succeeded in providing a modern and highly informative description of all the material traditionally covered in the context of a real and functional analysis course spread out over two semesters and intended for graduate students. The “Complements” sections that conclude each of the chapters as well as certain of the final chapters presenting results considered more advanced contain sufficient material to allow this text to be used as a reference manual for a more in-depth course that would extend over three semesters. However, it should be noted that the subject matter pushed back to these additional sections is compiled in a somewhat more disorderly fashion than the material that makes up the core of the text.

 

Real and Functional Analysis contains more than 500 problems, whose difficulty ranges from routine exercises (indicated with a ◦ ) intended to develop understanding and favour retention, up to the type of problems that may be part of the makeup of a PhD examination (those presenting a formidable level of difficulty are marked with an asterisk).

 

The makeup of the vast bibliography (that includes more than 700 titles) was obviously the subject of painstaking literature review. The authors have taken the time to include in the appendix a summary over several pages of the historical development of functional analysis and the various directions in which this discipline and its applications are used. There is then an orienting bibliography that will point the interested reader toward appropriate resources. 

 

Because this volume is part of Springer's Moscow Lectures series dedicated to the Moscow mathematical tradition, the choice of adopting the nomenclature that prevails in that school is self-evident. Attempts to finds the Cauchy–Schwarz inequality, the monotone convergence theorem or the Hahn–Banach separation theorem in the index will therefore be in vain. However, these results can certainly be found in the text, but respectively placed under the terms Cauchy–Bunyakovskii’s theorem, Beppo Levi’s theorem and theorem 6.3.7. That being said, since this work is intended for an international readership (indeed all signs indicate that it will appropriately meet the needs of graduate-level mathematics students around the world), it would nevertheless have been wise to enrich the index by also referencing the classical theorems under their commonly used names in the English-speaking world.

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Frederic Morneau-Guerin is a professor in the Department of Education at Universite TELUQ. He holds a Ph.D. in abstract harmonic analysis.