Abstract: 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, Dr. David Salinger, and Prof. Alex Strohmaier 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 [, , , ]. The other nice books with many interesting problems are [, ].

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
• Notations and Assumptions
  • Integrability conditions
• Motivating Example: Fourier Series
  • Fourier series: basic notions
  • The vibrating string
  • Historic: Joseph Fourier
• Basics of Metric Spaces
  • Metric Spaces
  • Useful properties of metric spaces
• Basics of Linear Spaces
  • Banach spaces (basic definitions only)
  • Hilbert spaces
  • Subspaces
  • Linear spans
• Orthogonality
  • Orthogonal System in Hilbert Space
  • Bessel’s inequality
  • The Riesz–Fischer theorem
  • Construction of Orthonormal Sequences
  • Orthogonal complements
• Duality of Linear Spaces
  • Dual space of a normed space
  • Self-duality of Hilbert space
• Fourier Analysis
  • Fourier series
  • Fejér’s theorem
  • Parseval’s formula
  • Some Application of Fourier Series
• Operators
  • Linear operators
  • Orthoprojections
  • B(H) as a Banach space (and even algebra)
  • Adjoints
  • Hermitian, unitary and normal operators
• Spectral Theory
  • The spectrum of an operator on a Hilbert space
  • The spectral radius formula
  • Spectrum of Special Operators
• Compactness
  • Compact operators
  • Hilbert–Schmidt operators
• Compact normal operators
  • Spectrum of normal operators
  • Compact normal operators
• Integral equations
• Banach and Normed Spaces
  • Normed spaces
  • Bounded linear operators
  • Dual Spaces
  • Hahn–Banach Theorem
  • C(X) Spaces
• Measure Theory
  • Basic Measure Theory
  • Extension of Measures
  • Complex-Valued Measures and Charges
  • Constructing Measures, Products
• Integration
  • Measurable functions
  • Lebesgue Integral
  • Properties of the Lebesgue Integral
  • Integration on Product Measures
  • Absolute Continuity of Measures
• Functional Spaces
  • Integrable Functions
  • Dense Subspaces in L_p
  • Continuous functions
  • Riesz Representation Theorem
• Fourier Transform
  • Convolutions on Commutative Groups
  • Characters of Commutative Groups
  • Fourier Transform on Commutative Groups
  • The Schwartz space of smooth rapidly decreasing functions
  • Fourier Integral
• Advances of Metric Spaces
  • The Stone–Weierstrass Theorem
  • Contraction mappings and fixed point theorems
  • The Baire Category Theorem and Applications
  • Semi-norms and locally convex topological vector spaces
• Tutorial Problems
  • Tutorial problems I
  • Tutorial problems II
  • Tutorial Problems III
  • Tutorial Problems IV
  • Tutorial Problems V
  • Tutorial Problems VI
  • Tutorial Problems VII
• Solutions of Tutorial Problems
  • Solution of Tuitorial Problem I
  • Solutions of Tutorial Problems II
  • Solutions of Tutorial Problems III
  • Solutions of Tutorial Problems IV
  • Solutions of Tutorial Problems V
  • Solutions of Tutorial Problems VI
  • Solutions of Tutorial Problems VII
• Course in the Nutshell
  • Some useful results and formulae (1)
  • Some useful results and formulae (2)
• Supplementary Sections
  • Reminder from Complex Analysis

This document was translated from L^AT_EX by H^EV^EA.