Advanced Functional Analysis and Integral Transforms

Functional analysis is a branch of mathematical analysis concerned with the study of infinite-dimensional vector spaces (primarily function spaces), their topologies, and the mappings between them. These mappings, known as operators, may act on infinite-dimensional spaces over the real or complex numbers, and include both operators and functionals. Functional analysis also studies properties of these mappings, such as duality, boundedness, compactness, continuity, measurability, and spectral theory, as well as their extensions to quantum algebras and C*-algebras.This book is a collection of notes, expository articles, talks, exercises, and lectures on advanced functional analysis and integral transforms, drawn from the seminar sessions held every Friday at Tecnológico de Estudios Superiores de Chalco. The present edition places particular emphasis on function representability and Fréchet spaces, which play an important role in the generalization of representations of infinite-dimensional groups through G-modules.