### Lectures on Functional Analysis by Prof. Abdumalik Rakhimov

09Feb11

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.

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.