With the rapid development of modern natural science and technology, higher requirements are put forward for the cultivation of engineering doctoral students. They should have a good modern mathematics accomplishment and the ability to study some nonlinear frontier or interdisciplinary scientific theories. In order to meet the needs of mathematics teaching for doctoral students majoring in non-mathematics, according to the requirements of the course syllabus and the specific situation of doctoral students in recent years, and on the basis of summing up many years of teaching experience, this textbook is carefully compiled, which aims at improving the cultivation of modern mathematics and learning the basic knowledge of nonlinear theory. This textbook is divided into four parts: The first part (chapter 1) is the foundation of modern analytical mathematics, which mainly includes the topological structure, algebraic structure and measure structure of abstract space and the basic knowledge of linear functional analysis. It not only contains the necessary preparatory knowledge for studying this course, but also provides reference for further study and research of doctoral students. ..