Clason C. Introduction to Functional Analysis

Springer, 2020. — 166 p. — (Compact Textbooks in Mathematics). — ISBN: 3030527832.Functional analysis has become one of the essential foundations of modern applied mathematics in the last decades, from the theory and numerical solution of differential equations, from optimization and probability theory to medical imaging and mathematical image processing.
This textbook offers a compact introduction to the theory and is designed to be used during one semester, fitting exactly 26 lectures of 90 minutes each. It ranges from the topological fundamentals recalled from basic lectures on real analysis to spectral theory in Hilbert spaces. Special attention is given to the central results on dual spaces and weak convergence.Metric Spaces
Compact Sets
Normed Vector Spaces
Linear Operators
The Uniform Boundedness Principle
Quotient Spaces
Linear Functionals and Dual Spaces
The Hahn–Banach Theorem
Adjoint Operators
Reflexivity
Weak Convergence
Compact Operators
The Fredholm Alternative
The Spectrum
Inner Products and Orthogonality
The Riesz Representation Theorem
Spectral Decomposition in Hilbert Spaces