Illustrative Introduction to Modern Analysis

Nikos Katzourakis is based at the University of Reading in the UK. His field of expertise lies in the analysis of nonlinear Partial Differential Equations and Calculus of Variations. His name is connected to contributions in vectorial variational problems for supremal functionals and generalised solutions for fully nonlinear systems.Eugen Varvaruca is based at the University "Al. I. Cuza’’ of Iasi in Romania. His research work is concerned with the analysis of nonlinear Partial Differential Equations, focusing particularly on free-boundary problems in fluid dynamics. In 2012 he was awarded the Whitehead Prize by the London Mathematical Society for his contributions.

• Sets, mappings, countability and choice. Metric spaces and normed spaces. Completeness and applications. Topological spaces and continuity. Compactness and sequential compactness. The Lebesgue measure on the Euclidean space. Measure theory on general spaces. The Lebesgue integration theory. The class of Lebesgue functional spaces. Inner product spaces and Hilbert spaces. Linear operators on normed spaces. Weak topologies on Banach spaces. Weak* topologies and compactness. Functional properties of the Lebesgue spaces. Solutions to the exercises.