ABOUT THE COURSE:Functional analysis is a branch of mathematical analysis that studies vector spaces with a limit structure (such as a norm or inner product), and functions or operators defined on these spaces. The aim of this course is to discuss with some basic concepts and abstract results of functional analysis, which are basic in nature but paramount importance from the point of view of applications indicating the variety of applications.INTENDED AUDIENCE: Under graduate/Post graduatePREREQUISITES: Knowledge of basic linear algebra and real analysis available on NPTEL portal is desirable.INDUSTRY SUPPORT: Functional Analysis is a core course for any mathematics discipline and having applications in many area of Science and Engineering. Also, Functional Analysis is an abstract branch of mathematics that originated from classical analysis.The impetus came from applications: problems related to ordinary and partial differentialequations, numerical analysis, calculus of variations, approximation theory, integral equations,and so on.
Overview
Syllabus
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 identityWeek 6:Orthogonality of vectors, Orthogonal complements, Gram- Schmidt orthonormalization process.Week 7:Bessel’s inequality, The conjugate space H*, Riesz representation theoremWeek 8:Operators on Hilbert spaces: The adjoint operator, Self adjint operatorWeek 9:Positive Operators, Normal Operators, Unitary OperatorsWeek 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.
Taught by
Prof. Akhilesh Prasad
