Week 1: Basic Analysis: Vector spaces, Linear transformation between vector spaces, Basic inequalities, Metric spaces and its properties. Convergence, Cauchy sequence, Completeness.
Week 2: Norms and Normed linear spaces: Definitions and properties, Convergence, Cauchy sequence, Completeness.
Week 3: Banach Spaces, Quotient spaces of Banach spaces, Illustrative examples.
Week 4: Bounded (continuous)linear operators, Bounded linear functionals, Dual spaces.
Week 5: Inner product spaces, Hilbert spaces,Illustrative examples of Hilbert spaces,Further properties of Inner product spaces, Schwartz’s inequality, Strong and Weak convergence, Applications of Polarization identity
Week 6: Orthogonality of vectors, Orthogonal complements, Gram- Schmidt orthonormalization process.
Week 7: Bessel’s inequality, The conjugate space H*, Riesz representation theorem
Week 8: Operators on Hilbert spaces: The adjoint operator, Self adjint operator
Week 9: Positive Operators, Normal Operators, Unitary Operators
Week 10: Partially ordered set, Zorn’s Lemma, Hahn-Banach Theorem (Normed spaces), Application to bounded linear functional on C[a,b].
Week 11: Baire’s category theorem, Uniform Boundedness principle and its applications.
Week 12: Open mapping theorem, Closed graph theorem. An application of the Banach’s theorem.
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