Syllabus
1. Measure Theory:
Sigma-algebras, measures, outer measures, completion,
construction and properties of the Lebesgue measure, non-measurable
sets, Measurable functions, point wise convergence, almost uniform
convergence, convergence in measure.
2. Integration: Lebesgue integration, limit theorems, comparison
with the Riemann integral, relationship with differentiation, functions
of bounded variation
and absolute continuity.
3. Signed Measures: Radon - Nikodym
theorem, Lebesgue decomposition theorem, change of variable formula,
Product Spaces, Fubini's theorem and
applications.
4. Lp-Spaces : Holder and Minkowski inequalities, completeness, convolutions, approximation by smooth functions, duality.
5.
Riesz representation
theorem: Riesz representation theorem for positive linear functionals,
Proof of the theorem, construction of the Lebesgue measure via this
approach.
- Teacher: Soumyashant Nayak
Syllabus
1. Topological Spaces: Topological
spaces, Bases, Continuous maps, Subspaces, Quotient spaces, Products,
Connectedness and Compactness.
2. Convergence: Nets, Filters, Limits; Convergence, Countability and Separation axioms.
3. Topological groups: Topological groups; Uniform structures, Products of Compact spaces; Compactifications.
4. Metrizability: Metrizability and Paracompactness, Complete Metric spaces and Function spaces.
5. Monodromy: Fundamental Group and Covering spaces.
https://www.isibang.ac.in/~adean/infsys/database/JRF/JTop1.html
- Teacher: Shreedhar Inamdar