{"id":593,"date":"2023-04-10T05:25:19","date_gmt":"2023-04-10T01:55:19","guid":{"rendered":"https:\/\/mohammadijoo.com\/blog\/?p=593"},"modified":"2023-04-10T06:05:13","modified_gmt":"2023-04-10T02:35:13","slug":"real-analysis-in-a-nutshell","status":"publish","type":"post","link":"https:\/\/mohammadijoo.ir\/en\/blog\/real-analysis-in-a-nutshell\/","title":{"rendered":"\u201cReal Analysis\u201d in a Nutshell"},"content":{"rendered":"\n<p style=\"text-align:justify\"> As a bachelor student in \u201cMathematics\u201d major or as a self-learner of mathematics major, in first semesters, you have to pass two mandatory courses: \u201cMathematical Analysis\u201d and \u201cReal Analysis\u201d.<\/p>\n\n\n\n<p style=\"text-align:justify\">  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 \u201cMathematical Proof\u201d, 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 \u201cReal Analysis\u201d course in universities.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p style=\"text-align:justify\"> This course usually starts with \u201cSet Theory\u201d and \u201cFunctions\u201d and their properties and every theorem or lemma needs to be proved during the course. There are lots of new definition you haven\u2019t heard about sets before, like: \u201cenumerable sets\u201d, \u201cdenumerable sets\u201d, \u201ccountable sets\u201d, \u201csupremum and Infimum of sets\u201d and etc.<\/p>\n\n\n\n<p style=\"text-align:justify\"> Then it will continue with \u201creal\u201d 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.<\/p>\n\n\n\n<p style=\"text-align:justify\"> The course continues with \u201cseries of real numbers\u201d. Then we have a definition for \u201cMetric Space\u201d and it defines what a \u201cMeter\u201d is. The \u201cCauchy series\u201d and its properties and related lemma are introduced and proved. We have \u201cContour Theorem\u201d for metric spaces.<\/p>\n\n\n\n<p style=\"text-align:justify\"> In next chapter \u201cMeasure Theory\u201d is introduced. \u201cAlgebra of sets\u201d is another topic which discussed in this course. Then it continues with \u201cintegral\u201d concept, but more advanced that what you know already about it. \u201cLebesgue integration\u201d is defined in this chapter. \u201cL\u00e9vy\u2013Steinitz theorem\u201d and \u201cFatou&#8217;s Lemma\u201d are introduced in this chapter. Another new concept about integrals is, \u201cRiemann Integral\u201d that is defined in this chapter as well.<\/p>\n\n\n\n<p style=\"text-align:justify\"> Next chapter is about \u201cNormed Vector Space\u201d or \u201cNormed Space\u201d. After definition of Normed Space, \u201cBanach Space\u201d and \u201cL<sub>P<\/sub> Space\u201d is defined. \u201cHolder Inequality\u201d, \u201cMinkowski Inequality\u201d and \u201cCauchy\u2013Schwarz inequality\u201d are three important properties in Normed Spaces. And at the end of this chapter \u201cHilbert Space\u201d and its related theorems are introduced.<\/p>\n\n\n\n<p style=\"text-align:justify\"> Below link is the outlines of \u201cReal Analysis\u201d course which is teaching in MIT university:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/ocw.mit.edu\/courses\/18-100c-real-analysis-fall-2012\/pages\/readings\/\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/ocw.mit.edu\/courses\/18-100c-real-analysis-fall-2012\/pages\/readings\/<\/a><\/p>\n\n\n\n<p style=\"text-align:justify\"> For Persian native speakers, there is very useful and organized course which covered almost all of related topics, prepared by \u201cFaraDars\u201d eLearning in below link:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/faradars.org\/courses\/fvtimth111-principles-of-real-analysis\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/faradars.org\/courses\/fvtimth111-principles-of-real-analysis<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>As a bachelor student in \u201cMathematics\u201d major or as a self-learner of mathematics major, in first semesters, you have to pass two mandatory courses: \u201cMathematical Analysis\u201d and \u201cReal Analysis\u201d. 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 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":598,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[102,130],"tags":[133,132,124,131],"class_list":["post-593","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-real-analysis","tag-banach-space","tag-hilbert-space","tag-mathematics","tag-real-analysis"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u201cReal Analysis\u201d in a Nutshell - Abolfazl Mohammadijoo&#039;s Blog<\/title>\n<meta name=\"description\" content=\"As a bachelor student in \u201cMathematics\u201d major or as a self-learner of mathematics major, in first semesters, you have to pass two mandatory courses:\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mohammadijoo.ir\/en\/blog\/real-analysis-in-a-nutshell\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u201cReal Analysis\u201d in a Nutshell - 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