Introduction to Mathematical Analysis

Notes

1. Uniformly converge (update): 0222
   Include: Continuous functional space & Abel’s theorem & Condition of derivative of sequence

2. Arzela-Ascoli Theorem: 0301
   Include: Heine–Borel theorem

3. Approximation Property: 0315
   Include: Volterra operator

4. Fubini’s Theorem: 0426
   Include: counterexample

Chap2-hint

Chap3-hint

A Note of Rabbe’s Test

A Note of Taylor Thoerem

Reference

1. W. Rudin, Principles of Mathematical Analysis.

2. Reference for basic leaner
   • R. Courant and F. John, Introduction to Calculus and Analysis
   • G. Folland, Advanced Calculus
3. Reference for topology
   • J. Lee, Introduction to Topological Manifolds
   • G. Bredon, Topology and Geometry
4. Reference for functional analysis
   • J. Conway, A Course in Functional Analysis
   • E. Kreyszig, Introductory Functional Analysis with Applications
   • B. Simon, Functional Analysis: Methods of Modern Mathematical Physics

Special Topics

1. Reference for probability theory
   • R. Vershynin, High-Dimensional Probability.
2. Reference for random process
   • J.R. Norris, Markov chains.
3. Reference for matrix computation
   • N. Higham, Functions of Matrices: Theory and Computation
   • K. Petersen and M. Pedersen, The Matrix Cookbook
4. Reference for fourier transformation (wavelet transformation)
   • S. Mallat, A Wavelet Tour of Signal Processing the Sparse Way
5. Reference for optimization method
   • S. Boyd and L. Vandenberghe, Convex Optimization.
   • G. Cala ore and L. Ghaoui, Optimization Models.