Analysis I

Description

This is the first part of an introduction to analysis in two volumes based on the author's undergraduate courses entitled Analysis I, II, and III, and the more advanced course entitled Tensor Analysis, at Heidelberg.

The contents of both Analysis I, and its sister volume, Analysis II, range from elementary calculus to fairly advanced topics in functional analysis, measure theory and differential geometry.

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 book can be used as a textbook, it comprises of materials for a one and a half semester course. Analysis I demands minimum prerequisites, and is intended as a textbook for first-year graduate students or for undergraduates who wish to graduate with a degree in Mathematics or Physics.

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