A course in mathematical analysis. Part 2. Book 1. Foundations of smooth analysis in multidimensional spaces. Theory of series. (Q2707795)

This is the third volume of a four-volume textbook covering a basic four-semester course ``Mathematical Analysis'' delivered by the author for the first and second years students at the Mathematical Department of the Novosibirsk State University. (See Zbl 1031.26001, Zbl 1031.26002 and Zbl 1031.26004 for reviews of the other three volumes.) One volume corresponds to one semester. The textbook is written in very detail and follows Russian traditions of teaching the calculus which goes back to [\textit{G.~M.~Fikhtengol'ts}, ``Foundations of analysis'', Vol. 1-3. (Russian) (1949; Zbl 0041.37807; 1948; Zbl 0033.10703; 1949; Zbl 0034.31903)]. On the other hand it represents the current state of teaching the mathematical analysis at the Novosibirsk State University: in the book under review, many subjects are treated from original points of view. We will give brief comments to the contents. NEWLINENEWLINENEWLINEChapter 9. Compact sets and topological spaces: \S 1. A survey of basic statements of Chapter 6. \S 2. A criterion for a set to be pre-compact. Lebesgue and Borel theorems. \S 3. The notion of a topological space. \S 4. Continuous mappings of topological spaces. In this chapter, starting from the following property of a closed interval \(I\subset\mathbb R\): ``for every sequence of points of \(I\), there exists its subsequence which converges to a point of \(I\)'', the author introduces the notions of compact and pre-compact sets, \(\varepsilon\)-net, and completely bounded set in a metric space. He establishes basic notions of continuous mappings of compact sets and, finally, extends the notions of continuous mapping and compact set to topological spaces. NEWLINENEWLINENEWLINEChapter 10. Fundamentals of non-smooth analysis: \S 1. General theorem on solvability of equations. \S 2. Inverse function theorem. \S 3. Consequences of the inverse function theorem. \S 4. Manifolds and systems of equations in \(\mathbb R^n\). \S 5. Conditional extrema. \S 6. Morse theorem. \S 7. Computing partial derivatives for implicitly defined functions. Examples. Using the contraction principle, the author obtains some sufficient conditions for invertibility of a smooth mapping. The latter are used to reach the main goal of this chapter, that is to prove an implicit function theorem. Several applications of the implicit function theorem, such as a theorem on the structure of a set of solutions of a system of equations and the Morse lemma about the structure of a \(C^2\)-smooth real function in a neighborhood of a non-degenerate critical point, are also discussed in this chapter. NEWLINENEWLINENEWLINEChapter 11. Theory of series: \S 1. Definitions. General statements on series. \S 2. Tests for a series to be convergent. \S 3. Abel and Dirichlet test for a series to be convergent. \S 4. The sum of values of a function over an arbitrary infinite set. \S 5. Infinite products. \S 6. Continued fractions. The author proves the Cauchy--Bolzano criterion for convergence of a series and establishes various tests for convergence of a series named after A.~L.~Cauchy, J.~d'Alembert, J.~L.~Raabe, P.~G.~Lejeune-Dirichlet, N.~H.~Abel, and G.~W.~Leibniz. The author also introduces the notion of the sum of values of a function over an arbitrary infinite set and applies it to the study of multiple series. He studies infinite products and given several conditions for them to be convergent. In particular, he proves the famous formula by J.~Wallis for \(\pi\). Besides, the author introduces continued fractions, studies their basic properties, and proves the convergence criterion by Ph.~L.~v.~Seidel. NEWLINENEWLINENEWLINEChapter 12. Functional series and integrals depending on a parameter: \S 1. The notion of uniform convergence for a family of functions. \S 2. Uniformly convergent functional series. \S 3. Power series. \S 4. Criteria for a function to be integrable over a closed interval. \S 5. Functions which can be represented as integrals depending on a parameter. \S 6. The Laplace method for constructing asymptotic representations. Stirling's formula. \S 7. Theorems about approximation of a function by polynomials. The notion of uniform convergence of functional series plays a crucial role in this chapter. Theorems on integration and differentiation of functional series are proven. As an illustration of the general theory developed, the Euler beta and gamma functions are introduced and their basic properties are proven. Besides, the Laplace method is described for asymptotic evaluation of an integral depending on a parameter and the well-known formula by J.~Stirling for \(n!\) is derived. NEWLINENEWLINENEWLINEEach chapter is followed by a list of exercises (181 exercises in total). Most of them are original ones and are selected in such a way as to provide students with the possibility of going deeper into the theories under discussion. NEWLINENEWLINENEWLINEThe textbook is intended for undergraduate students and university teachers.