It is well known that problem books and exercise collections play a special role in mastering theoretical material in functional analysis and the theory of functions of a real variable at pedagogical institutes and universities. In Russian, there exists a sufficient number of problem collections of various levels in this field. Among the most notable are the works of Yu.S. Ochan, A.A. Kirilov and A.D. Gvishiani, S.A. Telyakovsky, P.L. Ulyanov, A.N. Bakhvalov, M.I. Dyachenko, K.S. Kazaryan, P. Sifuentes, V.A. Petrov, N.Ya. Vilenkin, and M.I. Graev. These collections mainly focus on set theory and measure theory, and provide solutions to a large number of problems. However, the theory of integrals, operator theory, separability, completeness of spaces, compactness, and their applications to problem solving are insufficiently covered. In the Kazakh language, there are almost no problem collections corresponding to university curricula in these disciplines. Among the few published works, one may note G. Akyshev’s book on functional analysis, which covers several sections of the subject and includes solutions to problems, although its circulation is very limited. The aim of the study is to identify and systematize the methodological aspects of solving problems in the core areas of functional analysis, as well as to develop approaches to presenting them in the Kazakh language, taking into account the content of the university course. Since this discipline is one of the fundamental areas of mathematics, future researchers (Master’s and PhD students) cannot engage in scientific work without mastering it thoroughly. Functional analysis is considered a continuation of real analysis, and its problems are studied in close interconnection. From a methodological perspective, many problems were analyzed and their detailed solutions were provided. The study systematizes the main types of problems in set theory, measure theory, the Lebesgue integral, and functional analysis; defines methodological approaches to solving them; and develops a system of exercises of varying difficulty. The results obtained can be used in teaching functional analysis in the Kazakh language and in developing educational and methodological materials.