Troitsk: Troitsk branch of the state educational institution of higher professional education "Chelyabinsk State University", 2012. 86 s. 2nd Edition . , spr. and dop. This manual is the fourth part of the course Functional analysis and is devoted to the consideration of the theory and general properties of Hilbert spaces. The structure of the allowance repeats the previous parts. The manual presents theoretical and practical material on the topic "Hilbert spaces", studied by students of the specialty "Applied mathematics and computer science" . The manual is distinguished by a summary brevity and simplicity of presentation . The solution of the most complex problems is given as examples, to many problems for independent solution are given instructions. The training manual is intended for teachers and students. Can be used to conduct practical classes and organize independent work of students.