Jie Xiao 
Integral and Functional Analysis (Updated Edition) [PDF ebook] 

This textbook is based on three closely related courses: 1) Integration and Metric Spaces; 2) Lebesgue Integration; 3) Functional Analysis. Although the contents have been used for joint undergraduate and graduate courses, this textbook is designed primarily for senior undergraduate students. The prerequisites of this textbook are deliberately modest, and it is assumed that the students have some familiarity with calculus and linear algebra plus the basic (direct, indirect) proof methods.