###### *Real Analysis - Preface* ## Preface <small>*A course by Georg Grützner* ^[University of Luxembourg, Feb. 2025]^[Teaching Assistants: Alexey Balitskiy, Alexandre Benoist, Francesca Pistolato]</small> Welcome to **Analysis II**, a journey through some of the most profound and elegant ideas in mathematics. Building on the foundations laid in introductory analysis, this course explores deeper concepts that reveal the subtleties of limits, continuity, integration, and more. The topics in this text are not only essential for understanding advanced mathematics but also serve as the backbone of many applications in physics, engineering, and beyond. This course continues from [[RA - Course notes by Jean-Marc Schlenker.pdf|Analysis I of Jean-Marc Schlenker]]. It originates from the course notes of [Jean-Marc Schlenker](https://math.uni.lu/schlenker/teaching.html) on Analysis II, taught between 2016 and 2021 at the [University of Luxembourg](https://www.uni.lu/fstm-en/research-departments/department-of-mathematics/). Additionally, it draws inspiration from the works of [Stephen Abbott](https://link.springer.com/book/10.1007/978-1-4939-2712-8), [John B. Conway](https://www.cambridge.org/highereducation/books/first-course-in-analysis/482F2A3D79D2A3F7D1583AEE1C6A0506#overview), [Walter Rudin](https://en.wikipedia.org/wiki/Principles_of_Mathematical_Analysis), [Michael Struwe](https://people.math.ethz.ch/~struwe/skripten.html), and [Terence Tao](https://link.springer.com/book/10.1007/978-981-19-7284-3). Occasionally, I incorporated visually engaging videos from [3Blue1Brown](https://www.youtube.com/@3blue1brown), a YouTube channel created by [Grant Sanderson](https://www.3blue1brown.com/about). ![[32264483720_431fde2b4d_o.png]] In this course, we will focus on a small but significant corner of the vast and intricate map of mathematics: calculus. Along the way, we will occasionally glimpse its deep connections to other areas of mathematics. ^[The map of mathematics was created by [Dominic Walliman](https://dominicwalliman.com/). Visit his blog or YouTube channel to explore more of his beautifully crafted maps on science and mathematics.] Each chapter is crafted to build your intuition and technical skills while encouraging a rigorous approach to proofs and problem-solving. The content has been carefully organized to ensure a logical progression, with each topic naturally leading to the next. ### **How to Use This Book** This text is designed for first-year students who have completed an introductory course in analysis. It assumes familiarity with basic concepts like limits, continuity and differentiation in one variable. Each chapter builds incrementally, with definitions, theorems, and proofs presented alongside examples and exercises. **Tips for success:** 1. **Understand, don’t memorize.** The heart of analysis lies in understanding why results are true. Take the time to study proofs and reconstruct arguments on your own. 2. **Practice actively.** Work through the exercises, as they are designed to reinforce and extend your understanding. 3. **Ask questions.** Whether in class, study groups, or while reading, never hesitate to clarify doubts. Analysis thrives on curiosity. 4. **Be patient.** Some topics may initially seem abstract. Persist, and the beauty of these ideas will unfold.