IST Functional Analysis (2017)

M. A. Sofi, Department of Mathematics Kashmir University

• Lecture 1(23.10.2017): Resume on metric spaces, Motivation for studying functional analysis, Definition and examples of Banach spaces drawn from function spaces,sequence spaces, Bounded linear maps between normed linear spaces, their characterisations and examples.
• Lecture 2(24.10.2017): Functional analysis infinite dimensional spaces, Equivalence of norms and automatic continuity of linear maps on finite dimensional spaces, Local compactness as a characterisation of finite dimensional normed spaces.
• Lecture 3(25.10.2017): Heuristic motivation for fundamental theorems of functional analysis, Hahn Banach extension theorem and several of its applications.
• Lecture 4(26.10.2017): Closed graph theorem, inverse mapping theorem, the open mapping theorem and their applications.
• Lecture 5(27.10.2017): Projective topology induced by a family of mappings,Construction and examples, weak and weak* topology in Banach spaces as projective topologies, uniform boundedness principle and some of its applications.
• Lecture 6(28.10.2017): Banach Steinhauss theorem and applications, non-metrizability of weak and weak*-topology in infinite dimensional normed spaces,boundedness as a duality invariant, Equivalence of weak and norm-continuity of linear maps between normed spaces, Counterexamples to all the previous theorems in the absence of completeness of the spaces involved.
  

 Diganta Borah, Indian Institute of Science Education and Research, Pune

• Lecture 1 : October 23, 2017:Definition of Hilbert space, polarisation identity, definition of projection, optimisation techniques in Hilbert spaces.
• Lecture 1 : October 24, 2017:Definition of proximinal and Chebyshev normed linear spaces, orthogonal decomposition of Banach spaces, uniqueness of orthogonal decomposition in Hilbert spaces, Riesz representation theorem, definition of orthonormal basis of a Hilbert space.
• Lecture 1 : October 25, 2017:Existence of orthonormal basis for Hilbert spaces, Bessel’s inequality, introduction to fourier coeffients, Riemann-Lebesgue lemma, characterisation of an orthonormal basis.
• Lecture 1 : October 26, 2017:Theorem about rearrangements of series converging to the same limit in a Hilbert space, definition of separable Hilbert space, examples and calculation of orthonormal bases for various Hilbert spaces (Legendre polynomials, Hermite poly-
  nomials, Fourier basis).
• Lecture 1 : October 27, 2017:Definition of weak topology, discussion of convergence in weak topology, definition of weak∗topology, discussion of convergence in weak∗ topology, comparison of norm, weak and weak∗ topologies with examples, Banach-Alaoglu theo-
  rem.
• Lecture 1 : October 28, 2017:Definition and properties of adjoint of an operator, definition of spectrum, resolvant of an operator and various spectra (point, approximate, residual), computations of spectra for various familiar operators.

Vinayak Sholapurkar, Sir Parashurambhau College, Pune

• Lecture 1: 30-10-2017:Definition and examples of compact operators
• Lecture 2: 31-10-2017:Properties of compact operators .Approximation of Compact Operators on a Hilbert space by finite rank operators. Compact operator maps a weakly convergent sequence to a strongly convergence sequence.
• Lecture 3: 1-11-2017:Notion of spectrum of an operator and various spectral parts Some basic properties of spectrum and examples illustrating the nature of spectral parts .
• Lecture 4: 2-11-2017:Description of the spectrum of a compact operator and its characteristic properties.
• Lecture 5: 3-11-2017:Spectral theorem for compact self-adjoint operator .
• Lecture 6: 4-11-2017:Spectral theorem for compact normal operators.Notion of Fredholm alternative.

A. K. Vijayarajan, KSOM Kerala.

• Lecture 1; 30.10.2017: Adjoint of a bounded linear operator, properties and important results, Classification of bounded linear operators with respect to the adjoint.
• Lecture 2; 31.10.2017: Self-adjoint operators with an extensive discussion about projections. Examples
• Lecture 3; 1.11.2017: Normal and unitary operators and their norm characterisations. Spectrum of normal and unitary operators and related results.
• Lecture 4; 2.11.2017: Weak and strong operator topologies and the Double Commutant Theorem of von Neumann. Definition of von Neumann algebras.
• Lecture 5, 3.11.2017: Commutative C^∗-algebras and the the Gelfand-Naimark theorem. Unitazation of non-unital C^∗-algebras.
• Lecture 6; 4.11.2017: Representations and States of C and the GNS construction. Identification of categories of unital commutative C^∗-algebras with * -isometric isomorphisms and compact Hausdorff spaces with homeomorphisms.

 
Time Table

First Week : 23^rd October to 28^th October 2017

 AS : Amin Sofi ,DB : Diganta Bora ,GP : Geetanjali Phatak ,KM : Krishnat Masalkar

 Second Week : 30^th October to 4^th November 2017

 VR : Vijayrajan, VS : Vinayak Sholapurkar ,GP : Geetanjali Phatak KM : Krishnat Masalkar

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