Functional Analysis

This comprehensive introduction to functional analysis covers both the abstract theory and applications to spectral theory, the theory of partial differential equations, and quantum mechanics. It starts with the basic results of the subject and progresses towards a treatment of several advanced topics not commonly found in functional analysis textbooks, including Fredholm theory, form methods, boundary value problems, semigroup theory, trace formulas, and a mathematical treatment of states and observables in quantum mechanics. The book is accessible to graduate students with basic knowledge of topology, real and complex analysis, and measure theory. With carefully written out proofs, more than 300 problems, and appendices covering the prerequisites, this self-contained volume can be used as a text for various courses at the graduate level and as a reference text for researchers in the field.

這本全面介紹泛函分析的書籍，涵蓋了抽象理論以及在光譜理論、偏微分方程理論和量子力學中的應用。它從主題的基本結果開始，逐漸深入探討一些在泛函分析教科書中不常見的高級主題，包括Fredholm理論、形式方法、邊界值問題、半群理論、追蹤公式，以及量子力學中狀態和可觀測量的數學處理。本書對具備拓撲學、實分析、複分析和測度論基礎知識的研究生具有可讀性。書中詳細呈現了證明，提供了300多個問題，並附有前提知識的附錄，使得這本自成一體的書籍可以作為研究生課程的教材，也可以作為該領域研究人員的參考書。