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Analysis for applied mathematics. (English) Zbl 0984.46006

Graduate Texts in Mathematics. 208. New York, NY: Springer. viii, 444 p. (2001).
“Start with a pure mathematician, and turn him or her loose on real-world problems.” This is the author’s favorite algorithm for creating an applied mathematician, and this explains to some extent why the book under review is not a monograph about applied mathematics, but rather about topics like functional analysis, operator theory, Fourier analysis, approximation theory, mathematical physics, and measure theory, that impinge on the applied mathematical sciences.
The book consists of 8 chapters with the following headings: 1. Normed Linear Spaces; 2. Hilbert Spaces; 3. Calculus in Banach Spaces; 4. Basic Approximate Methods of Analysis; 5. Distributions; 6. The Fourier Transform; 7. Additional Topics (Fixed point theorems, compact and Fredholm operators, linear topological spaces etc.); 8. Measure and Integration. This enumeration shows that the choice of topics covered in the book is somewhat unusual: it seems that the author has collected some “appetizers” for students who do not have the time (and patience) to attend separate step-by-step lectures on functional analysis, mathematical physics, and Fourier theory. The reason for this may be the world-wide (highly deplorable) tendency to cut down high-school and university education in mathematics and to concentrate only on computer-based and application-oriented sciences.
Anyway, the book is very well written, and the presentation is clear and interesting. It is certainly a valuable complementation to the vast literature, a comparable work being E. Zeidler’s two volumes on applied functional analysis Berlin: Springer-Verlag (1995; Zbl 0834.46002 and Zbl 0834.46003)], and may be recommended as additional literature to graduate and PhD students in mathematics.

MSC:

46Bxx Normed linear spaces and Banach spaces; Banach lattices
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
32Wxx Differential operators in several variables
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
00A05 Mathematics in general
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
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