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Classical analysis on normed spaces. (English) Zbl 0839.46003
Singapore: World Scientific. xvi, 356 p. (1995).

This book provides an elementary but accelerated introduction to classical analysis on normed spaces with emphasis on nonlinear topics such as fixed points, calculus and ordinary differential equations. It assumes only general knowledge in finite-dimensional linear algebra, simple calculus and elementary complex analysis, since the rest of the treatment is self-contained.

Almost each paragraph contains some illustrative exercises (not very challenging, needing no hints) and each chapter ends with references to informative titles. The general references at the end of the book counts 165 titles.

The first three chapters contain the necessary background for any course in analysis: topology (for metric spaces) and Banach spaces (including Ascoli and Stone-Weierstrass theorems).

Chapter 4 is devoted to simplicial complexes and simplicial approximations, opening the way to simplicial homology (not covered in this book).

Chapter 5 deals with topological fixed point theory, Brouwer’s fixed point theorem being derived from the Borsuk-Ulam theorem. It ends with the theorem on invariance of domain.

The next two chapters contain standard topics of linear functional analysis, including Hahn-Banach and open mapping theorems. The weak topologies are not mentioned, weak convergence for sequences being used instead.

Vector valued maps of a scalar variable are introduced in chapter 8. The holomorphic functions are treated in the context of Banach spaces, with applications to the resolvent map.

The chapters 9, 10 are devoted to advanced calculus, including the implicit mapping theorem and Taylor’s formula in Banach spaces.

Chapter 11 deals with the initial value problem ${x}^{\text{'}}=f\left(t,z\right)$. Linear ODEs are treated via exponentiation not requiring Jordan forms. A generalized Peano theorem is presented in infinite-dimensional spaces.

The next three chapters deal with compact operators, the Fredholm alternative, operators in Hilbert spaces, including polar decomposition.

The last chapter (the 15th) is devoted to tensor products with some improvements due to the author.

The book is a valuable tool for a consistent course for a master degree, being written in a very accurate style.

##### MSC:
 46-01 Textbooks (functional analysis) 46G05 Derivatives, etc. (functional analysis) 47-01 Textbooks (operator theory) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces