Basic Analysis I is a textbook by Jiří Lebl that covers the fundamentals of real analysis, such as limits, continuity, differentiation, integration, and sequences and series. The book is suitable for undergraduate students and includes exercises and solutions. It is also available as a free PDF or a paperback on Amazon.
1 Introduction. We begin by discussing the motivation for real analysis, and especially for the reconsideration of the notion of integral and the invention of Lebesgue integration, which goes beyond the Riemannian integral familiar from clas- sical calculus. 1. Usefulness of analysis.
honours undergraduate-level real analysis sequence at the Univer-sity of California, Los Angeles, in 2003. Among the undergradu-ates here, real analysis was viewed as being one of the most dif-ﬂcult courses to learn, not only because of the abstract concepts being introduced for the ﬂrst time (e.g., topology, limits, mea-
From here, there are some very important deﬁnitions in real analysis. We say that b 0 is the least upper bound,orthesupremumofEif A) b 0 isanupperboundforEand B) ifbisanupperboundforEthenb 0 b: Wedenotethisasb 0 = supE. Similarly,wesaythatc 0 isthegreatestlowerbound,ortheinﬁnimumofEif A) c 0 isalowerboundforEand B) ifcisalowerboundforEthenc ...
found in a ﬁrst-year graduate course in real analysis. Although the presentation is based on a modern treatment of measure and integration, it has not lost sight of the fact that the theory of functions of one real variable is the core of the subject. It is assumed that the student has had a solid course in Advanced Calculus. Although
0.2. ABOUT ANALYSIS 7 0.2 About analysis Analysis is the branch of mathematics that deals with inequalities and limits. The present course deals with the most basic concepts in analysis. The goal of the course is to acquaint the reader with rigorous proofs in analysis and also to set a ﬁrm foundation for calculus of one variable (and several
Real Analysis Part I: MEASURE THEORY 1. Algebras of sets and σ-algebras For a subset A ⊂ X, the complement of A in X is written X −A. If the ambient space X is understood, in these notes we will sometimes write Ac for X −A. In the literature, the notation A′ is also used sometimes, and the textbook uses A˜ for the complement of A.