18.100A Real Analysis (Fall 2020, MIT OCW) | Mathematics

18.100A Real Analysis

18.100A Real Analysis (Fall 2020, MIT OCW). Instructor: Dr. Casey Rodriguez. This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an understanding and construction of proofs. (from ocw.mit.edu)

Sets, Set Operations and Mathematical Induction

Lecture 01 - Sets, Set Operations and Mathematical Induction

Lecture 02 - Cantor's Theory of Cardinality (Size)

Lecture 03 - Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property

Lecture 04 - The Characterization of the Real Numbers

Lecture 05 - The Archimedean Property, Density of the Rotations, and Absolute Value

Lecture 06 - The Uncountability of the Real Numbers

Lecture 07 - Convergent Sequences of Real Numbers

Lecture 08 - The Squeeze Theorem and Operations Involving Convergent Sequences

Lecture 09 - Limsup, Liminf, and the Bolzano-Weierstrass Theorem

Lecture 10 - The Completeness of the Real Numbers and Basic Properties of Infinite Series

Lecture 11 - Absolute Convergence and the Comparison Test for Series

Lecture 12 - The Ratio, Root, and Alternating Series Tests

Lecture 13

Lecture 14 - Limits of Functions in terms of Sequences and Continuity

Lecture 15 - The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function

Lecture 16 - The Min/Max Theorem and Bolzano's Intermediate Value Theorem

Lecture 17 - Uniform Continuity and the Definition of the Derivative

Lecture 18 - Weierstrass' Example of a Continuous and Nowhere Differentiable Function

Lecture 19 - Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem

Lecture 20 - Taylor's Theorem and the Definition of Riemann Sums

Lecture 21 - The Riemann Integral of a Continuous Function

Lecture 22 - Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula

Lecture 23 - Pointwise and Uniform Convergence of Sequences of Functions

Lecture 24 - Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits

Lecture 25 - Power Series and the Weierstrass Approximation Theorem

References

18.100A Real Analysis (Fall 2020)
Instructor: Dr. Casey Rodriguez. Lecture Notes and Readings. This course covers the fundamentals of mathematical analysis.