# Analysis, Fall 2014

Tuesday, Thursday 11:00-12:15 @ Warren Weaver Hall (WWH), Room 317

Instructor: Sivaram Ambikasaran, firstname@cims.nyu.edu
Office: WWH, 1105A. Office hours: Tuesdays 3:00 PM-5:00 PM or by appointment

Office: WWH, 1110. Office hours: Thursdays 4:00 PM-5:30 PM

Syllabus: This course is an introduction to rigorous analysis on the real line. Topics include: the real number system, sequences and series of numbers, functions of a real variable (continuity and differentiability), the Riemann integral, basic topological notions in a metric space, sequences and series of functions including Taylor and Fourier series.

Textbook:

Introduction to Real Analysis, 4th Edition, by Robert G. Bartle, Donald R. Sherbert

We will attempt to cover the entire book. Depending on time, we may omit Chapter 10 or 11 or both. I will also post notes if and when required.

Other recommended texts:

• Principles of mathematical analysis, by Rudin
• Elementary analysis: the theory of calculus, by Ross
• This course is a proof based course. Some useful texts on this front are:

• Mathematical Proofs: A Transition to Advanced Mathematics, by Gary Chartrand, Albert D. Polimeni and Ping Zhang
• The Nuts And Bolts of Proofs, by Antonella Cupillari
• For this course, we can get away with very basic naive set theory. For more details, below are some useful texts on this front:

• Naive Set Theory, by Paul R. Halmos
• Introduction to the theory of sets, by Joseph Breuer
• PLEASE READ THIS: It is most important that you regularly read the textbook and notes posted. You are also strongly encouraged to read the other recommended texts. Given that this is the first rigorous mathematics course for most of you, it is highly unlikely that you will be able to keep pace with the material presented without reading the textbook and notes. You are strongly encouraged to attend office hours. The most efficient way to learn this subject is to work out a lot of problems, problems, and more problems (exercises and examples from the book provide a wonderful opportunity) apart from homework problems. Getting a feel for definitions, theorems, etc will take time and this is where doing problems will help you to build your intuition on why things are defined in a specific way and why are certain theorems useful. Getting confused with analysis is a common phenomenon for most students. Spending sufficient time to understand a theorem is also common. Whenever you read a theorem, try to construct counterexamples (The textbook "Counterexamples in Analysis", by Bernard R. Gelbaum and John M. H. Olmsted is extremely useful on this front) by invaliding a particular assumption of the theorem. Some concepts are genuinely challenging and hard (especially since you will be learning it for the first time), but that's what makes this class entertaining and fun.

 Evaluation Assignments Quiz Midterm-I Midterm-II Final Exam Points 28 7 15 20 30

 Week Tuesday Thursday Homework 1 Motivation and § 1.1 (till Sets) §1.1 and §1.2 No recitation 2 § 1.2 and §1.3 § 1.3 and §2.1 HW1 HW1soln Q1 Q1soln 3 § 2.1 and §2.2 § 2.3, § 2.4 and §2.5 HW2 HW2soln Q2 Q2soln Proofs and problem solving 4 § 2.5 and § 3.1 § 3.1 and § 3.2 HW3 HW3soln Q3 Q3soln 5 § 3.2 and § 3.3 Monotone example Monotone sequence theorem § 3.3 and § 3.4 HW4 HW4soln No Quiz 6 MIDTERM 1 Midterm 1 solutions Midterm 1 statistics § 3.4 and § 3.5 HW5 HW5soln Q4 Q4soln 7 Fall break § 3.6 and § 3.7 HW6 HW6soln 8 § 3.7 and § 4.1 § 4.1 and § 4.2 HW7 HW7soln Q5 Q5soln ExtraQ ExtraQsoln 9 § 4.2 and § 5.1 § 5.2 and § 5.3 HW8 HW8soln Q6 Q6soln 10 § 5.3 and § 5.4 § 5.3 and § 5.6 HW9 HW9soln Q7 Q7soln 11 § 5.6 and § 6.1 MIDTERM 2 Midterm 2 solutions Midterm 2 statistics HW10 HW10soln Q8 Q8soln 12 § 6.2 and §6.3 § 6.3 and §6.4 HW11 HW11soln Q9 Q9soln 13 § 6.4 and §7.1 Thanksgiving Recess Thanksgiving Recess 14 § 7.1 and §7.2 § 7.2 and §7.3 HW12 HW12soln HW13 HW13soln Q10 Q10soln 15 §7.4, §8.1 and §8.2 §9.1, §9.2 and §9.3 HW14 HW14soln 16 Final exam solutions Final and course statistics

## Homework

There will be a total of 14 written homework. Homework will be posted on this website and will be due on Friday during recitations. Late homework will not be accepted. Graded homework will be returned to students during the recitations.

Grader(s) will be expecting you to express your ideas clearly, legibly, and completely, often requiring complete English sentences rather than merely just a long string of equations or unconnected mathematical expressions. This means you could lose points for poorly written proofs or answers. Clear exposition is a crucial ingredient of mathematical communication. Clarity of thought and presentation is more important in mathematics than any other field. The only way to master exposition is by repeated practicing. Read a proof from a book, then try to articulate and write the proof in your own words. Then proof read, imagining that someone else wrote the exposition and try to figure out, which parts need more work and are difficult to comprehend. You will be receiving feedback on your homework and you should use this feedback to present your work better the next time.

If the homework is not stapled, the grader is not responsible for losing your work.

## Quiz

There will be short (5-10 minutes) quiz each week either in class or during recitations.

## Exams

Midterm-I : October 7th, Tuesday in class.
Practice midterm I.
Practice midterm I solution.

Midterm-II : November 13th, Thursday in class.
Practice midterm II.
Practice midterm II solution.

Final exam : December 16th, Tuesday from 10AM - 11:50AM @ WWH 317.
Syllabus for the final exam.
Practice final.
Practice final solution.

There will be no make up exams. All exams will be closed book, closed notes. No internet/calculator allowed.

## Tentative Calendar

 Week Tuesday Thursday Friday (Recitation) 1 Sept 2: Motivation, Sets & Functions Sept 4: Sets & Functions, Mathematical Induction Sept 5: No recitation 2 Sept 9: Mathematical Induction, Finite and infinite sets Sept 11: Algebraic and order properties of $$\mathbb{R}$$, Absolute value Sept 12: HW1 due 3 Sept 16: Applications of supremum property, Intervals Sept 18: Sequences and their limits, Limit theorems Sept 19: HW2 due 4 Sept 23: Monotone sequences, Subsequences Sept 25: Limit points, limsup, liminf Sept 26: HW3 due 5 Sept 30: Bolzano-Weierstrass theorem, Cauchy Criterion Oct 2: Properly divergent sequences, Introduction to infinite series Oct 3: HW4 due 6 Oct 7: Midterm-I Oct 9: Limits of functions, Limit theorems Oct 10: HW5 due 7 Oct 14: Fall break Oct 16: Some extensions of the limit concept, Continuous functions Oct 17: HW6 due 8 Oct 21: Combinations of continuous functions, Continuous functions on intervals Oct 23: Uniform continuity, Monotone and inverse functions Oct 24: HW7 due 9 Oct 28: Derivative Oct 30: Mean value theorem Oct 31: HW8 due 10 Nov 4: L'Hospital's rules, Taylor's theorem Nov 6: Riemann integral and Riemann integrable functions Nov 7: HW9 due 11 Nov 11: The fundamental theorem of calculus Nov 13: Midterm-II Nov 14: HW10 due 12 Nov 18: Darboux integral Nov 20: Point wise and uniform convergence, Interchange of limits Nov 21: HW11 due 13 Nov 25: Exponential, logarithmic and trigonometric functions Nov 27: Thanksgiving Recess Nov 28: Thanksgiving Recess 14 Dec 2: Absolute convergence, tests for convergence Dec 4: Series of functions Dec 5: HW12 and HW13 due 15 Dec 9: Open and closed sets in $$\mathbb{R}$$, compact sets Dec 11: Continuous functions, Metric spaces Dec 12: HW 14 due and review