Analysis I (Graduate Series in Analysis)

This is the first part of an introduction to analysis in two volumes based on the author's undergraduate courses, Analysis I -- III, and the more advanced course, Tensor Analysis, at Heidelberg. The contents of both volumes range from elementary calculus to fairly advanced topics in functional analysis, measure theory and differential geometry. The first volume, Analysis I, covers some fundamental concepts of logic, set theory and the real numbers, the convergence of sequences and series in the real line, Euclidean spaces as well as Banach spaces, topological concepts including continuity, compactness and connectedness, differentiation in one variable, the theorems of Arzela-Ascoli and Stone-Weierstraß and analytic functions in several variables, as well as the Riemann integral. The present book can be used as a textbook, it comprises of materials for a one and a half semester course. The book, which demands minimum prerequisites, is intended as a textbook for first year graduate students or for undergraduates who later want to graduate in Mathematics or Physics.