Introduction
Functional Analysis is the study of vector spaces endowed with limits (topologies or norms) and the linear operators acting on them, extending classical linear algebra and calculus into infinite-dimensional settings. It provides the language and tools (such as Banach and Hilbert spaces, continuous linear operators, and spectral theory) to formulate and solve problems in differential equations, quantum mechanics, signal processing, and more.
Contents
Hilbert Spaces
Bounded Operators
Orthogonality and Bounded Linear Maps
Adjoints and Riesz Representation Theorem
Hilbert-Schmidt Operators
Compact Operators
Unbounded Operators
Unbounded Operators
Symmetric and Self-Adjoint Operators
The Spectral Theorem of Unbounded Operators
Stone’s Theorem
Nelson’s Counterexample
Nelson’s Commutator Theorem
Banach Spaces
Banach Spaces
The Implicit Function Theorem
The Hahn-Banach Theorem
The Baire Category Theorem