SF2713 Foundations of Analysis

During their first years at the university the students mainly encounter that part of analysis that was developed in the 17th and 18th centuries. The focus is on computations and very little time is spent on theoretical questions and proofs. The results are rather concrete and most students may feel that they are intuitively correct,or even self evident. So the need for proofs do not seem so urgent. But in the 19th century there was a revolutionary development in analysis, algebra and logic. Old concepts where abstracted and generalized and there was a greater need for rigorous proofs of their new highly non trivial results. It is this remarkable evolution of mathematics that this course reflects.

Aim of the course: To give the student a sufficient background in basic abstract analysis for further studies in subjects like functional analysis, Fourier analysis, differential equations, dynamical systems, representation theory, differential geometry, topology etc. Knowledge in these areas of mathematics is indispensable in understanding the highly non trivial use of abstract mathematics in the fascinating developments of todays theoretical physics.

Syllabus: Dedekind's cut, metric and normed spaces, topology, continuity, compactness, Banachs Fixed Point Theorem, inverse and implicit function theorem, Arsela-Ascoli Theorem, Stone-Weierstrass Theorem, differential forms, Stoke's Theorem, Lebesque Integrals.

Prerequisites: Analysis corresponding to SF1602 and SF1603 or SF1600 and SF1601 and preferably also complex analysis, differential equations and transforms corresponding to SF1628 and SF1629.

Requirements: Written examinations.

Required reading: Rudin, Walter, "Principles of Mathematical analysis".

Bonus system: There will be three written tests during the course that give at most 3+3+3 points in bonus for the final exam.

Examination: The final exam concists of eight tasks. Each of these gives at most three points. The grades are as follows:

A: 22-24; B: 19-21; C: 16-18; D: 14-15; E: 12-13; Fx: 11

Plan for the course: We will use Walter Rudin's classical book "Principles of Mathematical Analysis" Chapters 1, 2, 3, 4, 7, 9 (only pages 204-228), 10 and 11. Chapter 10 on differential forms and chapter 11 on Lebesque integrals are only included in the course as an orientation. There will be no questions on chapter 10 and 11 on the final exam, but there may be some questions about basic definitions in these chapters on the third bonus test.