MAT 3120 (Fall 2010)

Analysis III

Syllabus

The syllabus will be updated as the course progresses. Unless otherwise indicated, section numbers and exercises refer to the text A Course in Modern Analysis and its Applications by Graeme L. Cohen.

Date Text Sections Material Recommended Exercises*
Sep 9 2.1, 2.2 Metric spaces: definitions and examples Section 2.4: (1)–(5)
Sep 14 2.2, 1.10, 2.5 More examples, convergence in a metric space Section 2.4: (6)–(8), (10)–(12)
Sep 16 1.10, 2.5, 2.6 Convergence in a metric space, sequences of functions, complete metric spaces Section 2.9: (1), (2), (5)–(8)
Sep 21 2.6, 2.7, 2.8 Complete metric spaces, metric subspaces Section 2.9: (3), (4), (9)–(12), (14)
Sep 23 3.1, 3.2 Mappings between metric spaces, The Fixed Point Theorem Section 3.5: (3), (8), (10)
Sep 28 3.3, 4.1 Applications of The Fixed Point Theorem, compact sets Section 3.5: (2)–(4); Section 4.5: (1)–(5)
Sep 30 4.2, 4.3 Arzelà-Ascoli Theorem Section 4.5: (7)–(9)
Oct 4 4.3, 5.1 Approximation theory, topological spaces Section 5.7: (1), (2)
Oct 7 5.1, 5.2 The metric topology, closed sets Section 5.7: (4a), (5)
Oct 12 5.2 Closed sets, separable metric spaces Exercises in lecture notes only.
Oct 14 5.3, 5.5 Connectedness, compactness Section 5.7: (7), (8)
Oct 19 N/A Midterm exam (covers material up to including lecture of October 12) N/A
Oct 21 5.3 Compactness Section 5.7: (15)
Nov 2 5.4, 5.5 Continuity, path-connectedness Section 5.7: (9)
Nov 4 5.5 Path-connectedness Exercises in lecture notes only.
Nov 9 6.1 Completions, normed vector spaces Section 6.4: (1b), (2), (4), (12)
Nov 11 6.2, 6.5 Convergence in normed spaces, finite-dimensional normed spaces Section 6.4: (3), (5)–(8), (11); Section 6.10: (2)
Nov 16 6.5, 6.6 Student course evaluations. Finite-dimensional normed spaces, approximation theory. Section 6.10: (3), (4)
Nov 18 6.6 Weierstrass approximation. Section 6.10: (5), (12)–(15)
Nov 23 7.1, 7.2 Bounded linear mappings Section 7.5: (2), (3), (9), (10), (13)
Nov 25 7.2, 7.3 Bounded linear mappings, linear functionals Section 7.5: (6), (7), (12), (14)
Nov 30 8.1, 8.2 Hahn-Banach Theorem, inner product spaces Section 8.6: (1)–(10)
Dec 2 9.1, 9.2 Hilbert space and Fourier series Section 9.8: (1), (3), (4)
Dec 7 N/A Review N/A

* The exercises listed here are always in addition to the exercises written in the lecture notes.

Notes

Below are lecture notes for the course. They are a single file that will be updated after each lecture. Please keep in mind that these are notes that the professor writes for himself when preparing the lecture and should be complemented by a student's own class notes. Often more will be said in class than is in the notes. If you notice any errors in these notes (even small typos), please inform the professor. This helps him as well as your fellow students.