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Analysis I

Claus Gerhardt
International Press
Publication Date: 
Number of Pages: 
International Series in Analysis
[Reviewed by
Michael Berg
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Claus Gerhardt’s Analysis I sports a short bibliography (seven items) including four particularly suggestive titles, namely, Jean Dieudonné’s Foundations of Modern Analysis, Paul Halmos’ Naïve Set Theory, Herbert Enderton’s Elements of Set Theory, and Alfred Tarski’s Introduction to Logic and to the Methodology of the Deductive Sciences. These texts’ prominence in Gerhardt’s approach to analysis indicate that the book he presents to his readers is somewhat atypical. And so, indeed, it is.

Gerhardt’s two-volume effort grew from his Heidelberg lecture series, Analysis I–III, taking the student from almost nothing (i.e. the foundations of logic and set theory) to an introduction to submersed manifolds: from baby analysis to baby differential geometry. I have had occasion to cover some of the same material in a trio of courses at my university over the last few semesters (not contiguously). The material Gerhardt covers comprises a considerable superset of what I did, and so it should, since, unlike mine, his course is not pitched at the undergraduate level per se. He states, in fact, that his “textbook is intended for first year graduate students or for undergraduates who later want to graduate in mathematics or physics.” My audience consisted of senior undergraduates almost exclusively, but happily there are undeniable parallels, even given an undoubtedly greater level of intransigence among my students than among his: I think that even today the German university system is by no means as egalitarian as ours — for better or for worse.

All kvetching aside, the material in Analysis I, II (meant for three Heidelberg semesters, and by Gerhardt’s description always more than what he could reasonably cover), is entirely appropriate for a high level introduction to the indicated material. In rough terms, Analysis I will serve well for the upper division undergraduate sequence, and its successor, Analysis II, will then do excellent service as a text for beginning graduate level analysis, with functional analysis leading off. I guess this entails something of an inversion in the usual order of things: Gerhardt’s material on the Lebesgue integral only appears some four chapters later. However, this is certainly not a problem, given how Gerhardt develops his themes.

It also bears mentioning that Gerhardt starts off his first book with a longish section dealing with preliminaries on logic and set theory: this material is often covered in today’s ever-so-popular “transition to proofs” courses, so it can generally be dispensed with. Then the course can properly start off with Gerhardt’s thorough discussion of convergence (from R to Rn to Banach spaces), after which it’s the usual suspects: continuity, differentiation, function spaces, and Riemann integration. And with that we come to the close of Analysis I.

Analysis II then goes on to deal with, first, functional analysis, as already mentioned, and then with differentiation in Banach spaces, the inverse and implicit function theorems (after Banach’s fixed point theorem), ODE, Lebesgue integration, and finally, with a nod to geometry (and physics, I guess), tensor analysis and some manifold theory. With tensor analysis in the game, Gerhardt’s focus falls on Riemannian manifolds (or semi-Riemannian ones: relax the requirement that gij be positive definite).

I like this set of books a lot: they’re well-written and very accessible. One quibble is that the exercise sets, while entirely on target, are a bit on the short side: not very much choice for the instructor. But so be it: the student should just do them all, or nearly all, I guess. And this is of course the best way to go in any advanced undergraduate or beginning graduate course. Gerhardt has certainly hit the mark.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

  1. Chapter 0: Foundations (Elements of Logic, Elements of set theory, Cartesian Product, Functions and Relations, Natural and Real Numbers)
  2. Convergence (Convergence in R, Infinite series in R, Convergence in Rn, Metric spaces, Series in Banach spaces, Uniform convergence, Complex numbers)
  3. Continuity (Topological concepts, Continuous maps, Compactness, The Tietze-Urysohn extension theorem, Connectedness, Product spaces, Continuous linear maps, Semicontinuous functions)
  4. Differentiation in one Variable (Differentiable functions, The mean value theorem and its consequences, De L'Hospital's Rule, Differentiation of sequences of functions, The differential equation x' = Ax, The elementary functions, Polynomials, Taylor's formula)
  5. Spaces of continuous functions (Dini's theorem, Arzela-Ascoli Theorem, The Stone-Weierstraß Theorem, Analytic functions)
  6. Integration in one variable (The Riemann integral, Integration rules, Monotone and continuous functions are integrable, Fundamental theorem of calculus, Integral theorems and transformation rules, Integration of rational functions, Lebesgue's integrability criterion, Improper integrals, Parameter dependent integrals)