23UMATC44 Elements of Mathematical Analysis Syllabus:

23UMATC44 Elements of Mathematical Analysis Syllabus – Annamalai University UG Syllabus Regulation 2023-24

Objectives of the Course

• Identify and characterize sets and functions and Understand, test and analyze the convergence and divergence of sequences, series.
• Understand metric spaces with suitable examples

UNIT-I: Sets and Functions:

Sets and elements- Operations on sets- functions- real valued functions- equivalence – countability – real numbers- least upper bounds.

UNIT-II: Sequences of Real Numbers:

Definition of a sequence and subsequence – limit of a sequence – convergent sequences – divergent sequences – bounded sequences – monotone sequences

UNIT-III:

Operations on convergent sequences – operations on divergent sequences – limit superior and limit inferior- Cauchy sequences.

UNIT-IV: Series of Real Numbers:

Convergence and divergence – series with non –negative terms – alternating series – conditional convergence and absolute convergence – tests for absolute convergence.

UNIT-V: Limits and Metric Spaces:

Limit of a function on a real line – Metric spaces – Limits in metric spaces

Continuous Functions on Metric Spaces:

Functions continuous at a point on the real line – Functions continuous on a metric space.

Recommended Text

• Richard R. Goldberg, Methods of Real Analysis: Oxford and IBH Publishing, 2020.

Reference Books

• M. Apostol, Calculus (Vol. I), John Wiley and Sons (Asia) P. Ltd., 2002.
• R.G. Bartle and D. R Sherbert, Introduction to Real Analysis, John Wiley and Sons (Asia) P. Ltd., 2000.
• E. Fischer, Intermediate Real Analysis, Springer Verlag, 1983.
• 4. K.A. Ross, Elementary Analysis- The Theory of Calculus Series- Undergraduate Texts in Mathematics, Springer Verlag, 2003.