Functional Analysis with Applications

An overview of the basics of functional analysis covers : from normed, Banach and Hilbert spaces, Lebesgue integration and spaces, over linear operators and functionals, compact and self-adjoint operators, to small parameters and fixed point theory; with a strong focus on examples, exercises and practical problems, thus making it ideal as course material but also as a reference for self-study. Including a new chapter about the Stieltjes integral. The main aim of this new chapter is to introduce and explain the basic concepts of the Stieltjes integral and some of its properties. First the differentiation of monotone functions are discussed and then functions of finite variation and their properties are studied, along with Helly's selection principle. This is one of the first books devoted to the theory of Measure of Noncompactness, The Kronecker Index and The Brouwer Degree. It Provides a clear and well-organized treatment of the concept behind the development of mathematics as well as solution techniques, with many examples illustrated the main theory. This textbook stands out for the quality of its proofs, the clarity of its explanations, and the richness of its exercises, which include hints and solutions.