Komornik Vilmos. Lectures on Functional Analysis and the Lebesgue Integral

Springer, 2016. — 417 p.This book is based on lectures given by the author at the University of Strasbourg. Functional analysis is presented first, in a nontraditional way: we try to generalize some elementary theorems of plane geometry to spaces of arbitrary dimension.
This approach leads us to the basic notions and theorems in a natural way. The results are illustrated in the small lp spaces.The Lebesgue integral is treated next by following F. Riesz. Starting with two innocent-looking lemmas on step functions, the whole theory is developed in a surprisingly short and clear manner. His constructive definition of measurable functions quickly leads to optimal versions of the classical theorems of Fubini– Tonelli and Radon–Nikodým.These two parts are essentially independent of each other, and only basic topological results are used. In the last part, they are combined to study various function spaces of continuous and integrable functions.