# Need help choosing a good book on functional analysis.

The following is the syllabus of a course named "Functional Analysis" which I'm about to attend in the forthcoming semester. It is in fact a 4 credit course. I'm planning to self-study the topic in addition to attending the lectures given that I will be at least a week ahead if I start my independent work tomorrow. The following is the syllabus.

Syllabus: Complete metric spaces: Contraction mapping theorem and Baire’s category theorem.

Normed linear spaces: Finite and infinite dimensional spaces; convergence; completeness and compactness; linear operators and bounded linear operators; Uniform boundedness theorem; Hahn Banach theorem (without proof); Compact linear operators; Linear functional and bounded linear functional; Generalized functions; dual spaces; weak convergence; Space of bounded linear operators and bounded linear functionals; Convergence.

Inner product spaces: Inner products and properties; Orthogonal complements; direct sums; orthogonal sets and sequences.

Hilbert spaces: Properties; closest point theorem and applications; Bounded linear operators and bounded linear functionals on Hilbert spaces; Riesz representation theorem and Lax- Milligram theorem (without proofs); Adjoint, self adjoint, unitary and normal operators.

Applications: Differential equations, optimization, approximation theory, etc.

I need help choosing a suitable book to study this subject. I do have a copy of Rudin's Functional Analysis but I'm not sure if it is the best. Please help. Thanks.

• It depends on your preferred style, I liked Kolmogorov & Fomin's "Introductory Real Analysis". For me it hit the right balance between motivating examples, germane unpretentious discussion & comments and depth. – copper.hat Jul 12 '17 at 20:53
• There are many books on functional analysis and as @copper.hat mentions, it is down to personal preference which book is right for you. Introductory Functional Analysis with Applications by Kreyszig is a popular choice, especially for self-study. – AloneAndConfused Jul 13 '17 at 7:57
• I really liked Brezis "Functional Analysis, Sobolev Spaces, and Partial Differential Equations". It's really easy to read and has plenty of exercises and their solutions, which makes it nice for self study. It covers most (if not all) of the topics you have listed in detail in the first half of the book. – yousuf soliman Jul 13 '17 at 20:28