### 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.

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
- all continuous function space
- All bounded infinite sequences
- space
- 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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