MATH 4331 - Introduction to Real Analysis I & Math 4332 - Introduction to Real Analysis II - University of Houston
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MATH 4331 - Introduction to Real Analysis I & Math 4332 - Introduction to Real Analysis II

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

- MATH 4331 - Introduction to Real Analysis I -

Prerequisite: MATH 3333.

Course DescriptionStudy of metric spaces and real-valued functions. Topics include: convergence of sequence, continuity and uniform continuity of functions, sequences and series of functions, differentiation, and Riemann-Stieltjes integration.

Instructor's DescriptionThe above is the description in the official university catalogue which is slightly outdated.  Topics may vary depending on instructor's choice, and which topics are covered in the second semester MATH 4332.  For example all instructors will cover, in addition to most of the topics above, rigorous series convergence tests, normed and inner product spaces, uniform convergence of functions including polynomial approximation and power series, Baire category, the Arzela-Ascoli theorem, etc. Some instructors may cover some other topics such as  the proof of the inverse/implicit function theorem in n dimensions, or Lebesgue integration in place of Riemann-Stieltjes above. See also the longer PDF syllabus linked on this site, and the topics list for MATH 4332 below.

Possible Textbooks(Note that several of these texts begin with chapters devoted to material from the prerequisite class MATH 3333 and Calculus III, or material from MATH 3334. Students can review most of these topics on their own in the first weeks; excessive overlap should be avoided)

  • N. L. Carothers, Real analysis, Cambridge (has excellent problem sets)
  • K. Davidson and A. P. Donsig, Real Analysis with Real Applications. ISBN: 978-0130416476
  • W. Rudin, Principles of Mathematical Analysis 3rd edition, ISBN-13: ‎ 978-0070542358
  • T. Tao, Analysis II 3rd edition, ISBN-13: 978-9380250656


Grading: Please consult your instructor's syllabus regarding any and all grading guidelines.

*Note: This course  provides a solid introduction to deeper properties of the real numbers, continuous functions, differentiability and integration needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333 (our first course in analysis), including an introduction to the real numbers, basic properties of continuous, differentiable and integrable functions on the real line, and an ability to do epsilon-delta proofs.


- Math 4332 - Introduction to Real Analysis II -

Prerequisite: MATH 4331.

Course DescriptionFurther development and applications of concepts from MATH 4331. Topics may vary depending instructor's choice; possibilities include: Fourier series, point-set topology, measure theory, function spaces, or dynamical systems

Textbook: See the textbook list above for 4331.


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