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Math 645: Analysis I
Fall 2018 Graduate Course Syllabus

NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated to report any such activities to the Instructor.

Course Information

Course Description: This is the first part of the two-semester course that introduces an application-minded student to foundations and modern techniques of real analysis. Topics covered in this course include measure and integration theory, L^p spaces, integral inequalities, topological and metric spaces, Banach and Hilbert spaces, contraction mapping, duality, weak convergence, compactness.

Number of Credits: 3

Prerequisites: MATH 546 or departmental approval.

Course-Section and Instructors

Course-Section Instructor
Math 645-001 Professor C. Muratov

Office Hours for All Math Instructors: Fall 2018 Office Hours and Emails

Required Textbooks:

Title Analysis
Author Lieb and Loss
Edition 2nd
Publisher Academic Mathematical Society
ISBN # 978-0821827833

University-wide Withdrawal Date:The last day to withdraw with a w is Monday, November 12, 2018. It will be strictly enforced.

Other Texts


DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. DMS takes these policies very seriously and enforces them strictly.

Grading Policy: The final grade in this course will be determined as follows:

Homework 30%
Midterm Exam 30%
Final Exam 40%

Attendance Policy: Attendance at all classes will be recorded and is mandatory. Please make sure you read and fully understand the Math Department’s Attendance Policy. This policy will be strictly enforced.

Homework Policy: Homework will be assigned during class times and collected every couple of weeks.

Exams: There will be one midterm exam held in class during the semester and one comprehensive final exam. Exams are held on the following days:

Midterm Exam October 25, 2018
Final Exam Period December 15 - 21, 2018

The final exam will test your knowledge of all the course material taught in the entire course. Make sure you read and fully understand the Math Department's Examination Policy. This policy will be strictly enforced.

Makeup Exam Policy: To properly report your absence from a midterm or final exam, please review and follow the required steps under the DMS Examination Policy found here:

Cellular Phones: All cellular phones and other electronic devices must be switched off during all class times.

Additional Resources

Accommodation of Disabilities: Disability Support Services (DSS) offers long term and temporary accommodations for undergraduate, graduate and visiting students at NJIT.

If you are in need of accommodations due to a disability please contact Chantonette Lyles, Associate Director of Disability Support Services at 973-596-5417 or via email at The office is located in Fenster Hall, Room 260.  A Letter of Accommodation Eligibility from the Disability Support Services office authorizing your accommodations will be required.

For further information regarding self identification, the submission of medical documentation and additional support services provided please visit the Disability Support Services (DSS) website at:

Important Dates (See: Fall 2018 Academic Calendar, Registrar)

Date Day Event
September 4, 2018 T First Day of Classes
September 10, 2018 M Last Day to Add/Drop Classes
November 12, 2018 M Last Day to Withdraw
November 20, 2018 T Thursday Classes Meet
November 21, 2018 W Friday Classes Meet
November 22 - 25, 2018 R - Su Thanksgiving Recess
December 12, 2018 W Last Day of Classes
December 13 & 14, 2018 R & F Reading Days
December 15 - 21, 2018 Sa - F Final Exam Period

Course Outline

Week Topics
1 Review of advanced calculus
2 Introduction to measure theory, Monotone Class theorem
3 Uniqueness of measures, measurable functions, Lebsgue integral
4 Monotone convergence and dominated convergence theorems
5 Fatou's lemma, product measures, Fubini's theorem, other theorems
6 Definition of L^p spaces, convexity, Jensen's inequality
7 Hölder's and Minkowski's inequalities
8 MIDTERM (OCT. 25) completeness of L^p spaces
9 Continuous linear functionals, weak convergence, lower semicontinuity of the norms, uniform boundedness principle
10 Dual of L^p, Riesz representation theorem, convolutions, Young's inequality
11 Approximation by smooth functions, Banach-Alaoglu theorem for L^p
12 Introduction to function spaces: Hilbert, Banach, metric, topological spaces
13 Spaces of continuous functions, Hölder's spaces, Arzela-Ascoli theorem
14 Contraction mapping theorem and its applications
Final EXAM Week: DECEMBER 15 - 21, 2018

Updated by Professor C. Muratov - 8/26/2018
Department of Mathematical Sciences Course Syllabus, Fall 2018