As a branch of mathematical analysis, functional analysis brought with it a change of focus from the study of functions on the Euclid space to the analysis of abstract infinite-dimensional spaces, for example, Banach spaces and Hilbert spaces. As such it established a key framework for the development of modern analysis, and is particularly useful for the study of differential and integral equations.
This elective course will introduce the basic concepts of the functional analysis, such as the normed linear space, duality, weak convergence, and the Hilbert space. The basic knowledge of linear algebra and elementary analysis is required. The course will also discuss several fundamental and important theorems, for example, the Hahn-Banach theorem, the Riesz-Frechet representation theorem, and the fixed point theorems. These topics present the basic structure of this subject, and loom large in the body of mathematics. Finally, there will be a careful balance between the theories and the applications. The students, who complete this course, are expected to be prepared for the advanced Math courses at the graduate level and for the modern development of the research in the analysis.