Textbook in PDF format This book is a clear and structured introduction to functional analysis, designed for undergraduate students in the mathematical sciences. Based on third- and fourth-year lecture courses at Kyoto University, it guides readers through the foundational concepts of infinite-dimensional vector spaces and linear operators. The presentation closely follows the flow of actual lectures, preserving the atmosphere of classroom learning and making complex ideas more approachable. The first part of the book focuses on standard topics such as normed spaces, Banach and Hilbert spaces, and bounded linear operators. The second part introduces operator theory, including compact operators and the spectral theorem for self-adjoint operators—essential tools in mathematical physics and modern analysis. Supplemental topics like locally convex spaces are clearly marked for optional reading, making the book adaptable to different levels of preparation and interest. Rather than serve as a comprehensive reference, the book emphasizes problem-solving and conceptual understanding. Exercises—many drawn from real homework assignments—come with thoughtful hints to encourage independent thinking. An appendix covers essential background material in measure theory. Blending rigorous mathematics with the author's reflections on influential texts, Functional Analysis for Mathematical Sciences is both a practical learning guide and a tribute to the depth and beauty of the subject. Preface. About the Author. Notation. Basics of Banach Spaces. Basics of Hilbert Spaces. The Dual Spaces. Consequences of Completeness. Locally Convex Spaces. The Spectrum of Bounded Operators. Compact Operators on Banach Spaces. Detailed Accounts on Compact Operators on Hilbert Spaces. Spectral Decomposition of Bounded Self-Adjoint Operators. Unbounded Operators on Hilbert Spaces. Appendix: Miscellaneous Facts in Analysis. Hints of Problems and Exercises. Bibliography. Index