Functional Analysis

The advances in mathematics and theoretical physics from the start of the twentieth century brought about a need for a more general abstract framework for the study of differential operators, and the functional spaces where these operator act. While Lebesgue’s theory of integration gave us a precise description of the functions to be studied, it was the development of Hilbert spaces and Banach spaces that gave us the correct setting for the study of operators. In this course we will study Hilbert and Banach spaces, the various notions of convergence on those spaces, and the behaviour of bounded and unbounded operators defined therein. Connections to differential equations will be shown in examples.