Course Description
Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.
Course Content
- Completeness property. Countable and Uncountable.
- Metric spaces, Examples--$l_p, C[a, b]$, Limit, Open sets, Convergence of a sequence, Closed sets, Continuity.
- Complete metric space, Nested set theorem, Baire category theorem, Applications.
- Totally bounded, Characterizations of compactness, Finite intersection property, Continuous functions on compact sets, Uniform continuity.
- Characterizations of connectedness, Continuous functions on connected sets, Path connected.
- Definition and existence of integral, Fundamental theorem of calculus, Set of measure zero, Cantor set, Characterization of integrable functions.
- Point-wise and uniform convergence of sequenceof functions, Series of functions, Power series, Dini's theorem, Ascoli's theorem, Continuous function which is nowhere differentiable, Weierstrass approximation theorem.
Course Audience
MSc 2 years and who has done MTH101A.
Refs.
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N. L. Carothers, Real Analysis.
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R. R. Goldberg, Methods of Real Analysis.
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W. Rudin: Principles of Mathematical Analysis.
Evaluation.
- Quiz + homework -- 30%
- Mid-sem (announced) -- 30%
- End-sem (announced) -- 40%