| Date | Topics | Reading |
|---|
| Mon 19th Sep |
Introduction and overview: sets and mappings, countability |
Rudin 1.3; 2.1-2.4; 2.11 |
| Wed 21st Sep |
Equivalence relations, the rationals, field axioms |
Rudin 2.3; 1.12-1.16; 1.1 |
| Fri 23rd Sep |
Ordered sets, LUB property, ordered fields |
Rudin 1.1; 1.5-1.11; 1.17-1.18 |
| Mon 26th Sep |
The archimedean property and the density of the rationals |
Rudin 1.19-22 |
| Wed 28th Sep |
Cuts and construction of the reals |
Rudin p. 17-21 |
| Fri 30th Sep |
Countability, countability of rationals, uncountability of reals |
2.4-2.14 |
| Mon 3rd Oct |
Metric spaces |
Rudin 2.15-2.30 |
| Wed 5th Oct |
Open and closed sets |
As above |
| Fri 7th Oct |
Compactness |
Rudin 2.31-2.42 |
| Mon 10th Oct |
No class (Colombus Day) |
None |
| Wed 12th Oct |
Compactness; perfection; Heine Borel |
Rudin 2.36-2.43 |
| Fri 14th Oct |
Cantor; connectedness; sequences |
Rudin 2.44-3.4 |
| Mon 17th Oct |
Sequences; subsequences; examples. |
Rudin 3.5-3.14 |
| Wed 19th Oct |
Cauchy sequence; monotonicity; series |
Rudin 3.15-3.20 |
| Friday 21st Oct |
Series; cauchy criterion, comparison test |
Rudin 3.20-3.30 |
| Mon 25th Oct |
Examples; root test; ratio test |
Rudin 3.31-3.37 |
| Wed 27th Oct |
Alternating series; power series; absolute convergence |
Rudin 3.33-3.46 |
| Friday 29th Oct |
Absolute convergence; multiplication of series; rearangement of series |
Rudin 3.46-3.55 |
| Mon 31st Oct |
Vector spaces; examples; properties |
Axler p. 1-12 |
| Wed 2nd Nov |
Subspaces; direct sums |
Axler p. 13-18 |
| Fri 4th Nov |
Linear independence; span |
Axler p. 21-34 |
| Mon 7th Nov |
Basis; dimension |
Axler p. 21-34 |
| Wed 9th Nov |
Dual and quotient spaces |
Class notes |
| Fri 11th Nov |
No Class? (Veterans Day)? (Else catch up and introduction to tensors) |
None |
| Mon 14th Nov |
Linear transformations; null spaces; range |
Axler p. 37-52 |
| Wed 16th Nov |
Matrices; invertibility |
Axler p. 53-58 |
| Fri 18th Nov |
Fundamental Theorem of Algebra |
Axler p. 63-72 |
| Mon 21st Mov |
Invariant subspaces; eigenvalues; eigenvectors |
Axler p. 76-84 |
| Wed 23rd Nov |
Upper triangular and diagonal matrices; finding eigenvectors |
Axler p. 85-95 |
| Fri 25th Nov |
No class (Thanksgiving break - recover from eating all that turkey) |
None |
| Mon 28th Nov |
Inner Products; Cauchy-Schwarz and Pythagorus Theorem |
Axler p. 98-107 |
| Wed 29th Nov |
Orthonormal basis; Gram-Schmidt; minimization problems |
Axler p. 108-116 |
| Fri 2nd Dec |
Linear functionals; adjoints and self adjoint operators |
Axler p. 117-121, 128-132 |
| Mon 5th Dec |
Spectral Theorem |
Axler p. 132-137 |
| Wed 7th Dec |
Positive operators; isometries; polar decomposition |
Axler p. 144-157 |
| Fri 9th Dec |
Jordan Form |
TBA |
| Mon 12th Dec |
Jordan Form |
TBA |
| Wed 14th Dec |
Trace and Determinant |
Axler p. 213-235 |
| Fri 16th Dec |
Trace and Determinant |
As above. |
| Mon 19th Dec |
Last class: determinant and volume; motivation for next semester! |
Axler p. 236-243 |
| Next semester |
Single and multivariable calculus as found in Rudin Ch 4 onwards. |
Rudin and Spivak |