Functional Analysis

Functional Analysis is based on the lecture notes of a Functional Analysis course taught for many years by Sandro Zagatti at SISSA (the International School for Advanced Studies, Trieste). Faithfully transcribed and typeset after his passing, the book preserves the clarity, structure, and pedagogical precision that defined his teaching. The text provides a rigorous and complete introduction to the fundamental concepts and methods of functional analysis, guiding readers from the study of metric and normed spaces to the more advanced theory of Banach and Hilbert spaces. It develops systematically the key results that underlie modern analysis the Hahn Banach theorem, the Baire category theorem, compact and weak topologies, dual spaces, compact operators, and reflexivity together with the classical spaces of continuous functions, sequence spaces, and Lp spaces. Each topic is presented with careful attention to logical structure and with numerous worked examples that illustrate both abstract results and their applications. The exposition emphasizes conceptual understanding and the ability to reason independently, following a style that makes abstract notions accessible without compromising mathematical rigor. The volume also includes a rich collection of exercises, many drawn from Zagatti s original written exams, offering readers a way to consolidate and test their understanding. The problems often require combining ideas from different chapters, encouraging synthesis rather than rote application.

Intended primarily for graduate and advanced undergraduate students in mathematics and physics, the book is also a valuable reference for researchers and instructors seeking a compact and elegant presentation of the core principles of functional analysis. It can serve as both a self-contained text for a first course and a preparatory reference for further study in analysis, partial differential equations, or mathematical physics. By preserving the depth and lucidity of Zagatti s lectures, Functional Analysis stands as both a pedagogical tool and a tribute to his contribution to the SISSA mathematics community.