Lectures on Functional Analysis by Prof. Abdumalik Rakhimov


This is an announcement post:

The Laboratory of Computational Sciences & Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia will be organizing a series of lectures on Functional Analysis given by Research Fellow, Prof. Abdumalik Rakhimov every Thursday at 4pm.

Prof. Abdumalik Rakhimov

The lecture begins tomorrow as shown below:

Lectures on Functional Analysis (Lecture 1)
by Prof. Abdumalik Rakhimov (Research Fellow, INSPEM, UPM)
Date: Thursday, February 10, 2011
Time: 4.00-5.00pm
Venue: Al-Khawarizmi Seminar Room, Maths Bldg, UPM

The lectures cover the following topics:

Section 1: Metric Spaces

  • Basic concepts, definitions and examples 
  • n-dimensional Euclidean space
  • Cauchy-Schwarz inequality
  • Holder’s inequality
  • Holder’s integral inequality
  • Minkowski’s integral inequality
  • C[a,b] all continuous function space
  • All bounded infinite sequences
  • l_2 space
  • l_p space
  • Continuous mappings and homeomorphisms
  • Isometric spaces
  • Closure of a set, limit points
  • Convergence and limits
  • Dense subsets, separable spaces
  • Closed sets, examples
  • Open sets
  • Open and closed sets on the real line
  • The Cantor set
  • Complete metric space, definitions and examples
  • The nested sphere theorem
  • Baire’s theorem
  • Completion of a metric space
  • Definition of a contraction mapping, fixed point theorem
  • Contraction mappings and differenetial equations
  • Contraction mappings and integral equations

Section 2: Topological Spaces

  • Definitions and examples of topological spaces
  • Comparison of topologies
  • Convergent sequences in a topological space
  • Axioms of separation
  • Continuous mappings, homeomorphisms
  • Compact topological spaces
  • Compactness in metric spaces, total boundedness
  • Relative compact subsets of a metric space

Section 3: Linear Spaces

  • Definition and examples of linear spaces
  • Linear dependence
  • Linear subspaces
  • Factor spaces
  • Linear functionals
  • Normed linear spaces, definitions and examples
  • Subspaces of a normed linear space
  • Euclidean spaces, scalar products, orthogonality and bases
  • Existence of an orthogonal basis, orthogonalization
  • Bessel’s inequality, closed orthogonal systems
  • Complete Euclidean spaces, Riesz-Fischer theorem
  • Hilbert space, isomorphism theorem
  • Subspaces of Hilbert space, orthogonal complements and direct sums
  • Topological linear spaces, definitions and examples

Section 4: Linear Functionals

  • Continuous linear functionals on a topological linear space
  • Continuous linear functionals on a normed linear space
  • Hahn-Banach theorem for a normed linear space
  • Conjugate space of  topological linear space
  • Weak topology in topological linear space

Section 5: Remarks on Measure and Integration

  • Measure of elementary sets
  • Lebesgue measure of plane sets
  • Measurable functions, basic properties
  • Simple functions, algebraic properties on measurable functions
  • Equialent functions
  • Almost everywhere convergence
  • Lebesgue integral, definition and basic properties
  • Some properties of the Lebesgue integral
  • Lebesgue integral vs Riemann integral
  • Some spaces of integrable functions

All are invited and no registration fee.


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