As a bachelor student in “Mathematics” major or as a self-learner of mathematics major, in first semesters, you have to pass two mandatory courses: “Mathematical Analysis” and “Real Analysis”.
Many concepts and definitions introduced in these 2 courses, are familiar to you, as you learned in high school mathematics courses. But there is one important point in these 2 courses and that is practicing “Mathematical Proof”, because as a mathematician, you need to learn the proving skills as you need in rest of your scientific career as a mathematician. In this article, we cover the contents which is teaching in “Real Analysis” course in universities.
This course usually starts with “Set Theory” and “Functions” and their properties and every theorem or lemma needs to be proved during the course. There are lots of new definition you haven’t heard about sets before, like: “enumerable sets”, “denumerable sets”, “countable sets”, “supremum and Infimum of sets” and etc.
Then it will continue with “real” numbers. Before we continue, I remind again that during the course and each chapter, there are tens of new definitions about each concepts. At first, these definitions may seem useless and very abstract, but many of them will be used during other courses in rest of bachelor degree and also in advanced graduate courses in Master and PhD degrees.
The course continues with “series of real numbers”. Then we have a definition for “Metric Space” and it defines what a “Meter” is. The “Cauchy series” and its properties and related lemma are introduced and proved. We have “Contour Theorem” for metric spaces.
In next chapter “Measure Theory” is introduced. “Algebra of sets” is another topic which discussed in this course. Then it continues with “integral” concept, but more advanced that what you know already about it. “Lebesgue integration” is defined in this chapter. “Lévy–Steinitz theorem” and “Fatou’s Lemma” are introduced in this chapter. Another new concept about integrals is, “Riemann Integral” that is defined in this chapter as well.
Next chapter is about “Normed Vector Space” or “Normed Space”. After definition of Normed Space, “Banach Space” and “LP Space” is defined. “Holder Inequality”, “Minkowski Inequality” and “Cauchy–Schwarz inequality” are three important properties in Normed Spaces. And at the end of this chapter “Hilbert Space” and its related theorems are introduced.
Below link is the outlines of “Real Analysis” course which is teaching in MIT university:
https://ocw.mit.edu/courses/18-100c-real-analysis-fall-2012/pages/readings/
For Persian native speakers, there is very useful and organized course which covered almost all of related topics, prepared by “FaraDars” eLearning in below link:
https://faradars.org/courses/fvtimth111-principles-of-real-analysis