Real AnalysisⅠ

  • 申立勇
  • Created: 2014-12-08
Real AnalysisⅠ

 

Course No.S070100XJ010

Course Category basic course of subject      

Period/Credits40/2

PrerequisitesMathematical analysis, Real variable functions, Functional analysis, General topology.

Aims & Requirements

The curriculum is a core course for both the postgraduates and the doctorates from all fields of mathematics and also an elective course for graduates who major in theoretical physics.  The main contents of the course include the theory of Lebesgue integration on abstract measure spaces, the theory of  spaces, measure theory, differentiation of functions, integration on product spaces, transform theory such as Fourier transform, wavelet, Hilbert transform. Through learning the course, the students can master the basic concepts, methods, skills of modern real analysis and lay foundation for further studying modern mathematics and doing further research work in mathematics.

Primary Coverage

Chapter 1  Abstract integration 

Integrable functions; Properties of integration; The sequence of integrable functions and its convergence; Dominated convergence theoremFubini theorem;

Chapter 2  Positive Borel measures

σ-ring and σ-algebra generated by set family; The definition and properties of measure spaces; Lebesgue measure; Simple functions; Measurable functions; Product measure.

Chapter 3  Complex measures

Total variation; Absolute continuity; Consequences of Radon-Nikodym theorem; Bounded linear functional on ; The Riesz representation theorem.

Chapter 4  Differentiation

Derivatives of measures; The fundamental theorem of calculus; Differentiable transformation.

Chapter 5  Integration on product spaces

Measures on product spaces;  Product measures; Fubini theorem; Completion of product measures; Convolutions; Distribution functions.

Chapter 6  Fourier transforms

General properties of Fourier transforms; The inversion theorem; The Plancherel theorem; The Banach algebra .

Textbooks: W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., 1987.

References

1.  Paul R. Halmos, Measure Theory, GTM18, Springer-Verlag, New York, 1974.

2.  S, Lang, Real and Functional Analysis, GTM142, Springer-Verlag, New York, 1993.

Author::Dunyan YanMathematical Science, GUCAS                          

DateJune, 2009.