Introduction to Banach Spaces and Algebras

This graduate-level text offers a detailed exploration of functional analysis, focusing on Banach spaces and algebras, making it essential for advanced students.

In Introduction to Banach Spaces and Algebras, Graham Allan presents a comprehensive graduate-level text that delves into the intricate world of functional analysis, particularly focusing on Banach algebras. This subject is central to pure mathematics, and Allan's meticulous and clear approach makes the material accessible to final-year undergraduates and master's students. Drawing from his lectures at Cambridge University, the book assumes a foundational understanding of core mathematical concepts, such as metric spaces, analytic topology, and complex analysis, while not requiring prior knowledge of Lebesgue measure and integration theory.

The text begins with foundational concepts related to Banach spaces, including dual spaces and bounded linear operators. It systematically introduces key theorems that form the backbone of functional analysis, such as the Banach-Alaoglu theorem, Hahn-Banach theorem, and the closed graph theorem. The book also explores practical applications, including Fourier series and operators on Hilbert spaces, making it a valuable resource for those looking to apply theoretical concepts in real-world scenarios.

The latter part of Introduction to Banach Spaces and Algebras focuses on the theory of Banach algebras, featuring a thorough discussion on the holomorphic functional calculus in one and several variables. Allan provides essential background theory, supported by numerous examples and applications. To reinforce learning, exercises are included at the end of each section, while extensive notes guide readers to more advanced topics and resources. Edited posthumously by Professor H. G. Dales, this work is an indispensable tool for aspiring mathematicians.