How do you choose the best functional analysis book for your level and goals?

1 Know your prerequisites

Before you dive into functional analysis, you should have a solid background in linear algebra, real analysis, and complex analysis. These topics are essential for understanding the concepts and techniques of functional analysis. You should be familiar with vector spaces, matrices, norms, inner products, convergence, continuity, differentiation, integration, series, and complex numbers. If you need to refresh your knowledge or fill some gaps, you can find many books and online resources that cover these topics.

2 Choose your level

Functional analysis books can be divided into three levels: introductory, intermediate, and advanced. Depending on your current skills and objectives, you should choose the level that matches your needs. Introductory books are suitable for those just starting out and wanting to learn the fundamentals of functional analysis. These books typically address topics such as metric spaces, Banach spaces, Hilbert spaces, linear operators, bounded and unbounded operators, spectrum, compact operators, and applications. Examples of introductory books include Functional Analysis by Kreyszig, An Introduction to Functional Analysis by Rynne and Youngson, and Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras by Bachman and Narici. Intermediate books are suitable for those with some experience in functional analysis who want to further their knowledge and skills. These books usually discuss topics such as weak topologies, dual spaces, adjoint operators, Fredholm theory, spectral theory, self-adjoint operators, positive operators, and advanced applications. Examples of intermediate books include Functional Analysis by Rudin, Functional Analysis by Brezis, and Introduction to Functional Analysis by Taylor and Lay. Finally, advanced books are designed for researchers and experts aiming to explore the boundaries of functional analysis and its connections with other fields. These books typically cover subjects such as Banach algebras, C*-algebras, von Neumann algebras, operator algebras, operator spaces, operator theory, functional analysis on non-Archimedean fields, and cutting-edge applications. Examples of advanced books are A Course in Functional Analysis by Conway, Functional Analysis: An Advanced Course by Kato, and Non-Archimedean Functional Analysis by Schikhof.

3 Consider your interests

Functional analysis is a rich and diverse field that has many branches and applications. Depending on your interests and goals, you might want to focus on some aspects of functional analysis more than others. For example, if you are interested in quantum mechanics, you might want to read a book that emphasizes the role of Hilbert spaces and self-adjoint operators. If you are interested in harmonic analysis, you might want to read a book that explores the connections between functional analysis and Fourier analysis. If you are interested in algebraic geometry, you might want to read a book that relates functional analysis and sheaf theory. Some books that cater to specific interests are Functional Analysis and Quantum Mechanics by Teschl, Functional Analysis and Harmonic Analysis by Stein and Shakarchi, and Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk.