MATH-UA 325 Analysis

Term: Spring 2023
Instructor: Dr. Michal Shavit
Level: Undergraduate

Topics

The real number system, sequences and series of numbers, functions of a real variable (continuity and differentiability), the Riemann integral, basic topological notions in a metric space, sequences and series of functions including Taylor and Fourier series.

Description

Introduction to rigorous analysis on the real line.

Topics students should master by the end of the semester:

• Fundamental properties of real numbers
• Convergence of sequences and series
• Elementary topology in metric spaces (open, closed, compact)
• Rigorous definition and first properties of the derivative
• Rigorous definition and first properties of the integral
• Modes of convergence for sequences and series of functions

Skills students should acquire:

• Write routinely short proofs with a good mathematical formalism
• Use the classical tricks in analysis proofs (epsilon/3 argument, construction of a sequence, compactness, diagonal argument)
• Have a good intuition of the various concepts: be able to give typical examples (or counterexamples) for basic statements
• Manipulate rigorously the tools of integral and differential calculus