**Best Reference Books on Functional Analysis**, which are used by students of top universities, and colleges. This will help you choose the right book depending on if you are a beginner or an expert. Here is the complete list of

**Functional Analysis Books**with their authors, publishers, and an unbiased review of them as well as links to the Amazon website to directly purchase them. If permissible, you can also download the free PDF books on Functional Analysis below.

## 1. Functional Analysis

1."Introduction to Topology and Modern Analysis" by G F Simmons
“Introduction to Topology and Modern Analysis” Book Review: This book covers the mathematical aspects of continuity and linearity in great detail. It provides a comprehensive explanation of their basic definitions and explores their relationship to each other in terms of topology and modern analysis. This book is a valuable reference for students, researchers, and professionals alike.
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2."Functional Analysis" by G Bachman and L Narici
“Functional Analysis” Book Review: This book provides an in-depth introduction to inner-product spaces, normed and metric spaces, and topological spaces. It covers topics such as complete orthonormal sets, the Hahn-Banach theorem and its consequences, spectral notions, square roots, and spectral decomposition theorem, among others. Detailed proofs of theorems, definitions, index of symbols, and end-of-chapter exercises are also included. This book is ideal for students with background knowledge in linear algebra, advanced calculus, physics, and engineering.
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3."Introduction to Functional Analysis" by A E Taylor
“Introduction to Functional Analysis” Book Review: This book systematically analyzes the theory of normed linear spaces and linear mappings between such spaces, with separate chapters on Banach algebras, the Krein-Milman theorem, weak topologies and duality, equicontinuity, and the theory of Fredholm operators. Recent advances in functional analysis are also included, with a special focus on closed unbounded linear operators. The unifying power of the abstract linear-space is studied using problems of linear algebra, classical analysis, and differential and integral equations, with illustrations drawn from ordinary differential equations. This book is a valuable resource for graduate students of mathematics.
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4."Functional Analysis" by B V Limaye
“Functional Analysis” Book Review: This book is divided into two sections, introducing a distance structure on a linear space with additional features. The first section covers normed spaces, their completeness and continuous linear maps, including the theory of compact operators. The second section deals with Hilbert spaces and spectral theorems for compact self-adjoint operators. Numerous examples and problems of varying levels of difficulty are provided to illustrate abstract concepts and applications of results. The book also indicates the approximate construction of a solution, along with its existence and uniqueness.
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5."Introductory Functional Analysis with Applications" by Erwin Kreyszig
“Introductory Functional Analysis with Applications” Book Review: This book provides a comprehensive exploration of the practical applications of functional analysis to the study of natural sciences and mathematics. It focuses on fundamental concepts, principles, methods, and major applications of functional analysis, with numerous problems on Hilbert space theory and Banach spaces solved to provide a better understanding of the subject.
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6."Functional Analysis" by P K Jain
“Functional Analysis” Book Review: This book introduces the fundamentals of real analysis, linear algebra, and metric spaces, along with the uniform notations in the initial chapters. It then discusses normed and Banach spaces, bounded linear operators, and bounded linear functionals, followed by chapters on the theory and specific geometry of Hilbert spaces, functionals, and operators on Hilbert spaces. Separate chapters on spectral theory and Schauder bases are also included. This book is designed for senior undergraduate- and graduate-level students.
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7."Functional Analysis: A First Course" by Nair
“Functional Analysis” Book Review: This book provides a concise treatment of functional analysis using various figures and examples. It discusses linear algebra and linear space before introducing operators and some basic theorems of functional analysis. Advanced topics like dual space considerations, compact operators, and spectral theory of Banach and Hilbert space operators are also addressed. The book then applies the theorems to problems that arise while solving operator equations. This book is ideal for postgraduate students in Mathematics.
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8."Functional Analysis" by Walter Rudin
“Functional Analysis” Book Review: This book offers a modern perspective on functional analysis from the analysis and applied mathematics viewpoint. It covers advanced topics such as Lamonosov’s invariant subspace theorem, Kakutani’s fixed point theorem, and an ergodic theorem in detail. The text includes examples and exercises, making it suitable for graduate courses in functional analysis.
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9."Textbook of Functional Analysis: A Problem-Oriented Approach" by Krishnan V K
“Textbook of Functional Analysis” Book Review: This comprehensive book provides a thorough understanding of functional analysis and its fundamental theorems through solved numerical problems. The spectral properties of compact operators on Banach spaces and Hilbert spaces are also discussed in detail with corresponding practice problems. Furthermore, problems based on the square root of a positive operator and exercise sets are provided. This is an ideal book for postgraduate students pursuing courses in mathematics.
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10."Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext)" by Haim Brezis
“Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext)” Book Review: This book is aimed at students studying mathematics, physics, and engineering across various disciplines. The primary focus is on functional analysis (FA) and partial differential equations (PDEs), covering topics such as dimensional PDEs and modern theory representation. The text also explores the analysis of numerical schemes based on Fourier techniques, maximum principles, and Lax’s theorem. It is recommended for students with a solid foundation in real analysis and later delves into applied mathematics related to linear and nonlinear PDEs. The book provides numerous solved examples and practice exercises to aid in comprehension.
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## 2. Computational Functional Analysis

1."Computational Functional Analysis" by Ramon E Moore and Michael J Cloud
“Computational Functional Analysis” Book Review: This book covers advanced engineering analysis and modern control theory, with a focus on linear spaces, topological spaces, matrix spaces, normed linear spaces, Banach spaces, inner product spaces, Hilbert spaces, linear functional, types of convergence in functional spaces, and order relation in function spaces. It includes over 100 problems and exercises with complete solutions, as well as answers and tutorial hints for students. This book is a valuable resource for students studying applied functional analysis.
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2."An Introduction to Functional Analysis in Computational Mathematics" by V I Lebedev
“An Introduction to Functional Analysis in Computational Mathematics” Book Review: This book provides an introduction to function analysis and the methods used in computational mathematics. It covers functional spaces and problems in the theory of approximation, linear operators and functional, iteration methods for the solution of operator equations, the Newton method, partial Eigenvalue problem, spaces of linear operators, and linear spaces. The properties and equations are described in detail with proper explanations, and theorems and methods are provided with figures and descriptive explanations. This book is beneficial for students studying functional analysis.
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3."Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements (Texts in Applied Mathematics)" by B D Reddy
“Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements (Texts in Applied Mathematics)” Book Review: This book explains the basics of functional analysis, with a focus on boundary value problems, the finite element method, real analysis, and its applications. It provides detailed examples and problem statements for boundary value and finite element methods. This book is useful for graduate students in mathematics, physical science, and engineering.
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4."Functional Analysis and Applications" by imusti
“Functional Analysis and Applications” Book Review: This is a comprehensive guide written by imusti that covers the fundamental concepts of functional analysis and its applications in various fields. The book is divided into eight chapters, covering topics such as metric spaces, Hilbert spaces, Banach spaces, linear operators, and spectral theory. It also includes discussions on the theory of distributions and the Fourier transform. The author presents the material in a clear and concise manner, making it accessible to both students and professionals. This book is an ideal resource for anyone interested in functional analysis and its applications.
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5."Real and Functional Analysis (Mathematical Concepts and Methods in Science and Engineering)" by imusti
“Real and Functional Analysis (Mathematical Concepts and Methods in Science and Engineering)” Book Review: The book covers topics such as functional spaces, linear operators, iteration methods, the Newton method, partial Eigenvalue problem, spaces of linear operators, and linear spaces. The book provides detailed explanations, figures, and theorems, making it easy for students to understand the concepts. The book is a useful tool for students who want to learn about functional analysis and its applications.
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6."Introduction to Measure Theory and Functional Analysis" by Piermarco Cannarsa and Teresa D'Aprile
“Introduction to Measure Theory and Functional Analysis” Book Review: This book covers measure theory and function analysis, with a focus on measure spaces, integration, product measures, Hilbert spaces, Banach spaces, signed measures, absolute continuous functions, and set value functions. The book includes examples and exercises at the end of each unit, as well as references for further reading. The theorems and methods are described in detail, making it a valuable resource for graduate students in mathematics.
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7."Applications of Methods of Functional Analysis to Problems in Mechanics" by P Germain and B Nayroles
“Applications of Methods of Functional Analysis to Problems in Mechanics” Book Review: This book discusses the application of functional analysis to problems in mechanics, with various papers presented at a joint symposium of IUTAM/IMU in Marseille in 1975. Topics covered include quasi vibrational problems, the nonlinear problems of fluid mechanics, and elasto-plastic systems. The book includes 44 chapters and is useful for advanced mathematics and research students.
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8."Computational Text Analysis: For Functional Genomics and Bioinformatics" by Soumya Raychaudhuri
“Computational Text Analysis: For Functional Genomics and Bioinformatics” Book Review: This book covers important topics such as sequence analysis, gene expansion data, applications to proteomics, complex functions, and interactions. The book includes practical examples and modern experiments, making it useful for students and researchers in the field of computational biology, bioinformatics, and computer science.
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## 3. Nonlinear Functional Analysis

1."Nonlinear Functional Analysis: A First Course (Texts and Readings in Mathematics)" by S Kesavan
“Nonlinear Functional Analysis: A First Course (Texts and Readings in Mathematics)” Book Review: This is a comprehensive book on nonlinear analysis. The book explains abstract tools used in nonlinear analysis, their applications to semilinear elliptic boundary value problems, and results on weak derivatives. The book also contains chapter-end exercises, making it useful for an introductory course or seminar on nonlinear functional analysis.
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2."An Introduction to Nonlinear Functional Analysis and Elliptic Problems (Progress in Nonlinear Differential Equations and Their Applications)" by Antonio Ambrosetti and David Arcoya Álvarez
“An Introduction to Nonlinear Functional Analysis and Elliptic Problems (Progress in Nonlinear Differential Equations and Their Applications)” Book Review: This book introduces basic tools used in nonlinear analysis and their applications to semilinear elliptic boundary value problems. It covers various approaches that can be applied to different model cases and includes further results on weak derivatives. The book also provides chapter-end exercises and serves as a practical text for an introductory course or seminar on nonlinear functional analysis.
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3."Linear and Nonlinear Functional Analysis with Applications" by Philippe G Ciarlet
“Linear and Nonlinear Functional Analysis with Applications” Book Review: This book offers a complete introduction to both linear and nonlinear functional analysis, including complete proofs and illustrations with various applications to numerical analysis, optimization theory, and partial differential equations. It also covers foundational material, historical notes, and provides original references for exploring important results. It is intended for advanced undergraduates, graduate students, and researchers.
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4."Nonlinear Functional Analysis and Its Applications (Nato Science Series C:)" by S P Singh
“Nonlinear Functional Analysis and Its Applications (Nato Science Series C:)” Book Review: The book contains the proceedings of an institute, featuring lectures and contributed papers on the latest developments and future directions in the field. Topics covered in the book include degree and generalized degree theory, Hamiltonian systems, fixed-point theory, linear and nonlinear differential and partial differential equations, Nielsen numbers theory, and applications to dynamical systems and bifurcation theory. It also covers specific subjects such as Hamiltonian systems, minimax theory, heat equations, pendulum equations, and nonlinear boundary value problems. This book is a valuable resource for anyone interested in these areas of mathematics.
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5."Nonlinear Functional Analysis" by Klaus Deimling
“Nonlinear Functional Analysis” Book Review: This book provides extensive commentary, examples, and challenging exercises on the development of Brower degree and its applications, degree mappings for infinite dimensional spaces, and surveys of monotone and accretive mappings. It also explores inverse function theory, implicit function theory, and Newton’s methods.
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6."Nonlinear Functional Analysis (American Mathematics Society non-series title)" by Rajendra Akerkar
This is an ideal reference book that covers a wide range of topics in nonlinear functional analysis. The book starts with an introduction to Banach spaces and fixed-point theorems, followed by discussions on convexity, variational inequalities, and critical point theory. It also covers topics like topological degree theory, bifurcation theory, and the calculus of variations. The final chapters focus on applications to partial differential equations, such as elliptic and parabolic equations. The book is well-written, with clear explanations and numerous examples, making it a valuable resource for mathematicians and researchers interested in nonlinear analysis.
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7."Nonlinear Functional Analysis: 121 (Lecture Notes in Pure and Applied Mathematics)" by P S Milojevic
“Nonlinear Functional Analysis: 121 (Lecture Notes in Pure and Applied Mathematics)” Book Review: This is a comprehensive textbook that covers a wide range of topics in nonlinear functional analysis. It covers topics such as Banach spaces, fixed-point theorems, variational inequalities, and critical point theory. The book also includes chapters on topological degree theory, bifurcation theory, and the calculus of variations. The final chapters focus on applications of nonlinear analysis to partial differential equations. The book is written in a clear and concise manner, making it accessible to graduate students and researchers alike.
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8."Spectral Theory and Nonlinear Functional Analysis" by Julian Lopez-Gomez
“Spectral Theory and Nonlinear Functional Analysis” Book Review: The book discusses important issues in spectral theory and nonlinear functional analysis. It explores the structure of the set of zeroes of nonlinear operators and presents new methods for solving nonlinear equations in Banach spaces. The book explains the construction of an optimal algebraic/analytic invariant and its application in calculating the Leray-Schauder degree. Additionally, it discusses the general properties of solution sets with minimal use of topological tools. This book is an essential resource for those interested in the interplay between nonlinear functional analysis and spectral theory.
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9."Delay Equations: Functional-, Complex-, and Nonlinear Analysis (Applied Mathematical Sciences)" by Odo Diekmann and Hans-Otto Walther
“Delay Equations: Functional-, Complex-, and Nonlinear Analysis (Applied Mathematical Sciences)” Book Review: This book introduces the mathematical theory of infinite dimensional dynamical systems by focusing on delay differential equations, which are a simple yet rich class of examples. It includes detailed proofs and numerous exercises, suitable for self-study or graduate-level courses. This book can also serve as a reference for basic results, and provides a practical understanding of applied functional analysis and dynamical systems.
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10."Functional Analysis" by K Yoshida
“Functional Analysis” Book Review: This book begins with a presentation on set theory, topological spaces, measure spaces, and linear spaces, followed by a clear explanation of semi-norms, a general theory of Banach and Hilbert spaces, and the theory of generalized functions. It also covers analytical theory of semi-groups, ergodic theory, and diffusion theory, integration of the equation of evolution, weak topologies, and duality in locally convex spaces and nuclear spaces. The book highlights the applications of functional analysis in various fields of modern and classical analysis, making it useful for graduating students and researchers in pure and applied mathematics.
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11."First Course in Functional Analysis" by G Goffman and G Pedrick
“First Course in Functional Analysis” Book Review: This is a comprehensive introduction to the subject. The book covers the fundamental concepts and theorems of functional analysis, including normed linear spaces, Banach spaces, Hilbert spaces, linear operators, and spectral theory. It also includes topics such as the Hahn-Banach theorem, the closed graph theorem, the open mapping theorem, and the uniform boundedness principle. Each chapter concludes with numerous exercises to help readers deepen their understanding of the material.
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