LEADER 04602nam 22007575 450 001 9910799485303321 005 20240313210643.0 010 $a3-031-30832-8 024 7 $a10.1007/978-3-031-30832-1 035 $a(MiAaPQ)EBC31054873 035 $a(Au-PeEL)EBL31054873 035 $a(DE-He213)978-3-031-30832-1 035 $a(EXLCZ)9929516173600041 100 $a20240104d2023 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe Big Book of Real Analysis$b[electronic resource] $eFrom Numbers to Measures /$fby Syafiq Johar 205 $a1st ed. 2023. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2023. 215 $a1 online resource (950 pages) 311 08$aPrint version: Johar, Syafiq The Big Book of Real Analysis Cham : Springer,c2024 9783031308314 327 $aPreface -- 1. Logic and Sets -- 2. Integers -- 3. Construction of the Real Numbers -- 4. The Real Numbers -- 5. Real Sequences -- 6. Some Applications of Real Sequences -- 7. Real Series -- 8. Additional Topics in Real Series -- 9. Functions and Limits -- 10. Continuity -- 11. Function Sequences and Series -- 12. Power Series -- 13. Differentiation -- 14. Some Applications of Differentiation -- 15. Riemann and Darboux Integration -- 16. The Fundamental Theorem of Calculus -- 17. Taylor and MacLaurin Series -- 18. Introduction to Measure Theory -- 19. Lebesgue Integration -- 20. Double Integrals -- Solutions to the Exercises -- Bibliography -- Index. 330 $aThis book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the study of measures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study. 606 $aMathematics 606 $aMathematical analysis 606 $aSequences (Mathematics) 606 $aDifferential equations 606 $aMeasure theory 606 $aFunctions of real variables 606 $aCālcul$2thub 606 $aAnālisi matemātica$2thub 606 $aSuccessions (Matemātica)$2thub 606 $aMathematics 606 $aAnalysis 606 $aSequences, Series, Summability 606 $aDifferential Equations 606 $aMeasure and Integration 606 $aReal Functions 608 $aLlibres electrōnics$2thub 615 0$aMathematics. 615 0$aMathematical analysis. 615 0$aSequences (Mathematics). 615 0$aDifferential equations. 615 0$aMeasure theory. 615 0$aFunctions of real variables. 615 7$aCālcul 615 7$aAnālisi matemātica 615 7$aSuccessions (Matemātica) 615 14$aMathematics. 615 24$aAnalysis. 615 24$aSequences, Series, Summability. 615 24$aDifferential Equations. 615 24$aMeasure and Integration. 615 24$aReal Functions. 676 $a510 700 $aJohar$b Syafiq$01586365 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910799485303321 996 $aThe Big Book of Real Analysis$93872753 997 $aUNINA