REAL ANALYSIS
Paper Code:
MAT122
Credits:
5
Contact Hours:
75.00
Max. Marks:
100.00
Unit I:
I
15.00
Algebra and algebras of sets, Algebras generated by a class of subsets, Borel sets, Lebesgue measure of sets of real numbers, Measurability and Measure of a set, Existence of Non-measurable sets, Measurable functions.
Unit II:
II
15.00
Realization of non-negative measurable function as limit of an increasing sequence of simple functions. Structure of measurable functions, Convergence in measure, Egoroff's theorem.
Unit III:
III
15.00
Weierstrass's theorem on the approximation of continuous function by polynomials, Lebesgue integral of bounded measurable functions, Lebesgue theorem on the passage to the limit under the integral sign for bounded measurable functions.
Unit IV:
IV
15.00
Summable functions, Space of square summable functions. Fourier series and coefficients, Parseval's identity, Riesz-Fisher Theorem.
Unit V:
V
15.00
Lebesgue integration on R2. Lp-spaces, Holder-Minkowski inequalities. Completeness of Lp-spaces.
Essential Readings:
- Shanti Narayan, A course of Mathematical Analysis ,S.Chand and Co. New Delhi.
- T.M.Apostol , Mathematical Analysis , Norosa Publishing House, New Delhi.
- Real Analysis , R.R.Goldberg, Oxford and IBH publishing Company, New Delhi.
- Real Analysis , Walter Rudin,
References:
- Jain and Kaushik, An introduction to Real Analysis , S.Chand and Co., New Delhi
- P.K.Jain
Academic Year:
