Elementary Functional Analysis. Georgi E. In this introductory work on mathematical analysis, the noted mathematician Georgi E. Shilov begins with an extensive and important chapter on the basic structures of mathematical analysis: linear spaces, metric spaces, normed linear spaces, Hilbert spaces, and normed algebras. The standard models for all these spaces are sets of functions hence the term "functional analysis" , rather than sets of points in a finite-dimensional space. Chapter 2 is devoted to differential equations, and contains the basic theorems on existence and uniqueness of solutions of ordinary differential equations for functions taking values in a Banach space.
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Silverman Translator. Introductory text covers basic structures of mathematical analysis linear spaces, metric spaces, normed linear spaces, etc.
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Book:Georgi E. Shilov/Elementary Functional Analysis
Elementary Functional Analysis