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.