Analysis I (Fall 2017)

Lectures

03/08 Abstract measure spaces: finite, σ-finite, semi finite measures: definitions and examples. Continuity of measures w.r.t. decreasing subsets. Proof of completeness of Lebesgue measure. Exercise: Subadditivity of outer measure, construction of Cantor ternary function, non-Lebesgue measurable set, and incompleteness of Borel measure on R.
04/08 Completion and saturation of measures. Borel measure and Baire measure. Exercise: Show that saturation of an infinite measure space cannot produce finite measure. Does the same statement hold for the σ-finite case?
07/08 Caratheodory Theorem, semi-algebra, algebra generated by a semi-algebra, sufficient conditions for the extension to a measure.
10/08 Construction of product measure, Measurability properties of sections of sets from Rσδ.
11/08 Measurable functions. Properties of measurable functions on abstract measure space. Integration of non-negative function w.r.t. an abstract measure. Fatou's lemma. Show that supremum of an uncountable collection of measurable functions need not be measurable.
14/08 Integration of an integrable function w.r.t. an abstract measure. Convergence of integrals.
17/08 Fubini's Theorem for indicator function of a Rσδ set with finite measure.
18/08 General Fubini's theorem.
21/08 Definition of topologically bounded and σ-bounded sets; outer-, inner- and quasi- regularity of Baire measures; topologically regular outer measure. Measurability of Borel sets w.r.t. topologically regular outer measure.
24/08 Sufficient condition for a non-negative map defined on a topology to induce a topologically regular outer measure. Existence and uniqueness of complete and saturated quasi(inner) regular Borel measure agreeing to a Baire measure on σ-bounded Baire sets. Exercise: Find an example of a compact set of a topological space which is not a Gδ.
28/08 Completing the previous theorem. Statement of Riesz-Markov Theorem.
31/08 Proof of Riesz-Markov Theorem.
01/09 Tutorial and Class test.
04/09 Riesz Representation Theorem for C(X), the set of all continuous real functions on a compact T2 space X.
07/09 Completing the proof.
08/09 L^p(μ) - the classical Banach spaces. Proof of completeness.
11/09 Hölder's inequality and its implication, Hölder conjugate. The action of L^qon L^p.
14/09 Riesz representation theorem of L^p(μ) for p finite and μ a σ-finite measure.
Continuation for one extra hour.
15/09 Uniform boundedness principle of bounded linear operators. Norm on the space of bounded linear operators from a normed linear space to another.
18/09 Tutorial. No class on 21/09, 22/09, 25/09 due to Midsem. No class on 28/09, 29/09, 02/10 due to Festival break.
25/09 Midsem
04/10 Hahn-Banach extension Theorem [by Shubham Namdeo & Namrata]. extra class.
05/10 Application of Extension Theorem, proof of "L¹ is not reflexive". [by Vishakh & Saikat].
Continuation for one half extra hour. Open Mapping Theorem.
06/10 Banach separation Theorem [by Prasun]. Closed Graph Theorem.
09/10 Hilbert Spaces, Complete orthonormal system in a separable Hilbert space, Fourier expansion, Parseval's theorem. Riesz representation theorem for separable Hilbert space. [by Subham Tripathy]
11/10 Definition of Spectrum of an operator and its types. Definition and Properties of the compact linear operators, extra class.
12/10 Riesz-Schauder theory of the spectrum of compact linear operators.
Continuation for one half extra hour. Exercise: Reading assignment from [6], Theorem12.1 for bounded inverse Theorem and Example 18.7 for an example of a compact operator having zero as non-eigenspectrum.
13/10 Definition and properties of the adjoint of a bounded linear operator on a Hilbert space, Definition of normal, unitary and self-adjoint operators, the norm of self-adjoint operators. A necessary and sufficient condition for normal or unitary operators, statement of spectral theorem for compact self adjoint-operators. Exercise: Reading assignment from [6], Theorem 23.1c, 24.8, 25.2, 25.3, 25.5, 26.2. Using Theorem 23.1c, prove that if perp of a subspace S (i.e. S^⊥) of H is trivial, then S is dense in H.
14/10 Proof of spectral theorem for compact self adjoint-operators. extra class. 19/10 is a Holiday, also no class on 16/10, and 20/10 due to my absence.
23/10 Class Test.
26/10 Integration in polar coordinates.
27/10 Holomorphic functions, Cauchy-Goursat Theorem, and Cauchy’s integral formula.
30/10 Some conditions satisfied only by constant holomorphic functions. Maximum modulus theorem.
02/11 Higher Derivatives and Morera’s theorem. [by Shubham De & Garima]
03/11 Schwarz lemma, Harmonic Function, and Poisson's formula. No class on 06/11
09/11 Poisson integral and Schwarz reflection principle.
Continuation for one extra hour.
10/11 Removable Singularity and Taylor's Theorem. A holomorphic function with all derivatives zero at a point is constant on its domain of definition.
13/11 Zeros and singularities, the residue formula. The argument principle and applications: Rouche’s theorem. Open mapping theorem,
16/11 Tutorial27/11 Endsem

Not covered: the automorphisms of the disc and the upper half plane.