Analysis for Applied Mathematics
This book evolved from a course at our university for beginning graduate stu dents in mathematics-particularly students who intended to specialize in ap plied mathematics. The content of the course made it attractive to other math ematics students and to graduate students from other disciplines such as en gineering, physics, and computer science. Since the course was designed for two semesters duration, many topics could be included and dealt with in de tail. Chapters 1 through 6 reflect roughly the actual nature of the course, as it was taught over a number of years. The content of the course was dictated by a syllabus governing our preliminary Ph. D. examinations in the subject of ap plied mathematics. That syllabus, in turn, expressed a consensus of the faculty members involved in the applied mathematics program within our department. The text in its present manifestation is my interpretation of that syllabus: my colleagues are blameless for whatever flaws are present and for any inadvertent deviations from the syllabus. The book contains two additional chapters having important material not included in the course: Chapter 8, on measure and integration, is for the ben efit of readers who want a concise presentation of that subject, and Chapter 7 contains some topics closely allied, but peripheral, to the principal thrust of the course. This arrangement of the material deserves some explanation.
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algebra assume axioms Banach space boundary-value problem bounded linear operator Cauchy sequence closed set compact operator compact set complete contains continuous functions continuous map converges convex set Corollary countable cr-algebra defined definition denote dense derivative differential equation disjoint distribution domain eigenvalues element equivalent Example exists finite finite-dimensional fixed point follows formula Fourier transform function f Hence Hermitian Hilbert space hypotheses inequality inner product inner-product space integral equation interval invertible Lebesgue measurable Lemma Let f linear map linear transformation matrix measurable functions measurable sets measure space method metric space multi-index neighborhood nonempty nonnegative normed linear space normed space notation open set orthogonal outer measure polynomial preceding theorem Proof Prove real numbers satisfies Section sequence xn Show solution solve subset subspace Suppose surjective test function theory topological space topology uniformly unique unit ball vector space write xn+i zero