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Title Real Analysis (Code-09050501)

Real Analysis (Code-09050501)
B.Sc. (Hons) – 5th semester
SYLLABUS

S.No.

Topic

Domain

Hours as per UGC

 

1

  • Riemann integral, integrability of continuous and monotonic function,
  • The Fundamental theorem of integral calculus,
  • Mean value theorems of integral calculus,

Must know

15

 

 

2

  • Improper integral and their convergence,
  • Comparison test, Abel’s test and Dirichlet’s test,
  • Frullani’s integrals
  • Integral as a function of parameters
  • Continuity, Differentiablity and integrability of an integral of a function of parameter.

Must know

15

 

 

 

3

  • Definition and examples of Metric spaces,
  • Neighborhoods, limit points, Interior points,
  • Open and closed sets, closure and interior, boundary points,
  • Subspace of a metric space, equivalent metrics,
  • Cauchy sequences, Completeness,
  • Cantor’s intersection theorem,
  • Baire’s Category theorem, contraction principle.

Must know

15

 

 

4

  • Continuous functions, uniform continuity,
  • Compactness for metric spaces,
  • Sequential compactness, Bolzano- Weierstrass property,
  • Total boundedness, Finite intersection property,
  • Continuity in a relation with compactness, connectedness, components, continuity in relation with connectedness

Nice to know

15

Books Recommended;

  •  P.K. Jain and Khalil Ahmed: Metric spaces, 2nd Ed., Narosa, 2004
  • T.M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi, 1985
  • R.R. Goldberg : Real Analysis Oxford & IBH publishing Co., New Delhi, 1970
  • D. Somasundaram and B. Choudhary : A First Course in Mathematicl Analysis, Narosa Publishing House,  New Delhi, 1997
  • Shanti Narayan : A Course of Mathematical Analysis, S. Chand & Co., New Delhi
Admission 2017