MTH631 Highlighted Handouts Set 4

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About the instructor

Dr. Malik did his MS and PhD (Mathematics) from University of La Rochelle, La Rochelle, France in 2009 and 2012, respectively. Prior to MS and PhD, Dr. Malik completed his MPhil and MSc (Mathematics) from Department of Mathematics, University of the Punjab, Lahore, Pakistan. He has been a liated with several universities in Pakistan and abroad. He has the experience of teaching a wide range of mathematics courses at undergraduate and graduate level.

Dr. Malik has published several research articles in international journals and conferences. His area of research includes the study of dierential equations with nonlocal operators and their applications to image processing. He is also interested in inverse problems related to reaction-diusion equations with nonlocal integrodierential operators and boundary conditions. These models have numerous applications in anomalous diusion/transport, biomedical imaging and non-destructive testing.

About the handouts

The books followed during this course are: W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976. ISBN: 9780070542358. and Consequently, the most of the examples considered in these notes are from the above mentioned books and their exercises, but not restricted to those books only. If you nd any typing error in the text kindly report to me by writing an email to iii

Course Information

Title and Course Code:

Number of Credit Hours: 3 credits

Course Objective: Real Analysis II is the follow up course of Real Analysis I and in general an advanced course related to mathematical analysis. The topics of the Real Analysis II are linked with its rst course namely Real Analysis I, indeed, we will extend the ideas of Real Analysis I to Euclidean space Rn, we will discuss sequences and series of functions, limits and continuity of functions of several variables, partial derivatives their applications, multiple integrals etc. Upon completion of this course students will be able to

• Understand the convergence of sequence of functions (LO1).
• Understand the pointwise convergence, uniform convergence, several tests for convergence (LO2).
• Apply the interchange of limit and integration, derivative of sequence of func- tions (LO3).
• Understand the in nite series of functions, convergence, Weierstrass's test and some other results about the convergence (LO4).
• Apply Dirichlet's test for uniform convergence, series of product of two func- tions, interchange of sum and intgeration (LO5).
• Represent and study the function which could be written as power series,term by term integral and derivative of a power series, (LO6). item Understand the concept of equicontinuous function, The Stone-Weierstrass Theorem (LO7).
• Understand and nd the Fourier series, Fourier coe cients, convergence of Fourier series (LO8).
• Apply the best approximation theorem and understand the Euler gamma func- tion and the beta function and their properties (LO9)
• Understand the functions of several variables, Heine-Borel Theorem, limits and continuity of functions of several variables (LO10)
• Vector valued functions and their calculus, Bounded functions and several results about vector valued functions (LO11)
• Dierentiablity in Rn, Dierentials, Directional derivatives, Partial deriva- tives, Maxima and minima (LO12)
• Improper integrals, Multiple integrals, Functions of bounded variation (LO15) iv

Prerequisites: Real Analysis I (MTH621)

The textbooks for this course:

[1] W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976. ISBN: 9780070542358.

[3] S. Ponnusamy, Foundations of Mathematical Analysis, Birkhauser, 2012.

Reference books:

[4] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Revised English Edition Translated and Edited by R. A. Silverman, Dover Publication, Inc. New York.

[5] R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, Third Edition, 2000, John Wiley & Sons Inc.