A First Course in Functional Analysis
Rabindranath Sen
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486 pages. Temps de lecture estimé 6h04min.
A comprehensive introduction to functional analysis, starting from the fundamentals and extending into theory and applications across multiple disciplines.‘A First Course in Functional Analysis: Theory and Applications’ provides a comprehensive introduction to functional analysis, beginning with the fundamentals and extending into theory and applications. The volume starts with an introduction to sets and metric spaces and the notions of convergence, completeness and compactness, and continues to a detailed treatment of normed linear spaces and Hilbert spaces. The reader is then introduced to linear operators and functionals, the Hahn-Banach theorem on linear bounded functionals, conjugate spaces and adjoint operators, and the space of linear bounded functionals. Further topics include the closed graph theorem, the open mapping theorem, linear operator theory including unbounded operators, spectral theory, and a brief introduction to the Lebesgue measure. The cornerstone of the book lies in the motivation for the development of these theories, and applications that illustrate the theories in action.
One of the many strengths of this book is its detailed discussion of the theory of compact linear operators and their relationship to singular operators. Applications in optimal control theory, variational problems, wavelet analysis and dynamical systems are highlighted.
This volume strikes an ideal balance between concision of mathematical exposition and offering complete explanatory materials and careful step-by-step instructions. It will serve as a ready reference not only for students of mathematics, but also students of physics, applied mathematics, statistics and engineering.One of the many strengths of the book is the detailed discussion of the theory of compact linear operators and their relationship to singular operators. Applications in optimal control theory, variational problems, wavelet analysis, and dynamical systems are highlighted.
This volume strikes the ideal balance between concision of mathematical exposition, and complete explanatory material accompanied by careful step-by-step instructions intended to serve as a ready reference not only for students of mathematics, but also students of physics, applied mathematics, statistics and engineering.Introduction; I. Preliminaries; II. Normed Linear Spaces; III. Hilbert Space; IV. Linear Operators; V. Linear Functionals; VI. Space of Bounded Linear Functionals; VII. Closed Graph Theorem and Its Consequences; VIII. Compact Operators on Normed Linear Spaces; IX. Elements of Spectral Theory of Self-Adjoint Operators in Hilbert Spaces; X. Measure and Integration Lp Spaces; XI. Unbounded Linear Operators; XII. The Hahn-Banach Theorem and Optimization Problems; XIII. Variational Problems; XIV. The Wavelet Analysis; XV. Dynamical Systems; List of Symbols; Bibliography; Index
