### MA 403   Real Analysis

• #### Course Timings:

• For M.Sc. (Mathematics) and M.Sc. (ASI) Students:
Mon   10.35 AM - 12.30 PM,   Tue   11.35 AM - 12.30 PM,   Thu   10.35 AM - 11.30 AM,
in Room 216, Department of Mathematics

• For Minor Area Students in the B.Tech. Programme:
Wed   9.30 AM - 10.55 AM,   Thu   5.05 PM - 6.05 PM,   Fri   9.30 AM - 10.55 AM,
in Room 105, Department of Mathematics
• #### Office Hours:

By appointment (Room 106 B, Maths Dept)

• #### Homeworks and Exams:

• Exercise Set 1 [Given: Jul 23-24]:   PDF File
• Exercise Set 2 [Given: Aug 12-13]:   PDF File
• Quiz 1   [Monday, Aug 18]:   PDF File
• Exercise Set 3 [Given: Aug 20-21]:   PDF File
• Exercise Set 4 [Given: Sep 1-2]:   PDF File
• Mid-Sem Exam:   [Sunday, Sep 7]:   PDF File
• Exercise Set 5 [Given: Sep 18]:   PDF File
• Quiz 2   [Tuesday, Oct 7]:   PDF File
• End-Sem Exam:   [Saturday, Nov 22]:   PDF File

• Notes on Uniform Convergence by Prof. B. V. Limaye:   PDF File

• #### Syllabus :

Review of basic concepts of real numbers: Archimedean property, Completeness.

Metric spaces, compactness, connectedness, (with emphasis on Rn).

Continuity and uniform continuity.

Monotonic functions, Functions of bounded variation; Absolutely continuous functions. Derivatives of functions and Taylor's theorem.

Riemann integral and its properties, characterization of Riemann integrable functions. Improper integrals, Gamma functions.

Sequences and series of functions, uniform convergence and its relation to continuity, differentiation and integration. Fourier series, pointwise convergence, Fejer's theorem, Weierstrass approximation theorem.

• #### Text/References:

S. R. Ghorpade and B. V. Limaye, A Course in Calculus and Real Analysis, Springer, New York, 2006. [Second Indian Reprint, Springr (India), 2007.]

W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1983.