Introduction to Functional Analysis

This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington and Dr. David Salinger used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more. The notes are available also for download in PDF. The suggested textbooks are [1,6,8,9]. The other nice books with many interesting problems are [3, 7]. Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them! CONTENTS List of Figures 3 Notations and Assumptions 4 Integrability conditions 4 1. Motivating Example: Fourier Series 4 1.1. Fourier series: basic notions 4 1.2. The vibrating string 8 1.3. Historic: Joseph Fourier 10 2. Basics of Linear Spaces 11 2.1. Banach spaces (basic definitions only) 12 2.2. Hilbert spaces 14 2.3. Subspaces 16 2.4. Linear spans 19 3. Orthogonality 20 3.1. Orthogonal System in Hilbert Space 21 3.2. Bessel’s inequality 23 3.3. The Riesz–Fischer theorem 25 3.4. Construction of Orthonormal Sequences 26 3.5. Orthogonal complements 28 4. Fourier Analysis 29 Date: 16th October 2017. 1