This textbook discusses all useful topics in functional analysis and applications. It contains complete definitions of topics well-explained by suitable examples, explanations, and proofs throughout the book. The book studies major topics on normed linear spaces, bounded linear functionals, Banach spaces, Hahn-Banach theorem, reflexive spaces, open mapping theorem, closed graph theorem, Banach-Steinhaus theorem, spaces of bounded linear functionals, weak and weak* topologies, Hilbert spaces, Riesz representation theorem, Lax-Milgram lemma, Banach algebra, the spectrum and the Gelfand-Mazur theorem, commutative Banach algebra, differentiation and integration in Banach spaces, variational methods and optimizations, approximation and optimization, operator theory, nonlinear operators and applications, spectral theory, Sturm-Liouvillesystem and Fredholm alternative, fixed point theory, Banach contraction principle, Brouwer’s fixed point theorem, Schauder’s fixed point theorem, variational inequalities and applications, frames and bases in Hilbert spaces, frames and bases in Banach spaces
This book is designed as a textbook for a two-semester course in functional analysis, as well as for a first-semester course in functional analysis at the lower undergraduate level. The book is also accessible to junior mathematics majors who have studied advanced calculus or multivariable calculus. The essential prerequisites for reading this book are quite minimal: not much more than a stiff course in analysis, advanced calculus, and abstract algebra.