Linear Analysis

Math 675-001 (Fall 2024)

Detailed Syllabus

The following table contains the tentative schedule for the course. It will be updated regularly throughout the semester. The Pages column refers to the pages in the notes that are covered each week. The last column indicates the corresponding sections in the book by Kolmogorov and Fomin, referred to as [KF], which was used to define the preliminary exam syllabus. Throughout the semester, I will also add relevant references to the book by Kreyszig using [K], which in my opinion is a more accessible text. Further book recommendations for additional reading can be found at the end of this page.

Week	Dates	Pages		Book Sections
1	08/26 - 08/30	1-17	I. Introduction
			1. Superposition principle
			2. Solvability conditions
2	09/02		No class! (Labor Day)
			II. Metric Spaces
	09/03 - 09/06	18-22	1. Basic definitions	[KF] 5.1, [K] 1.1, 1.2
3	09/09 - 09/13	23-35	2. Convergence and continuity	[KF] 5.2, 6.2, [K] 1.4
4	09/16 - 09/20	36-47	3. Topological properties	[KF] 6.1, 6.4, 6.5, 6.6, [K] 1.3
			4. Separable metric spaces	[KF] 7.1, 7.2, 7.4, [K] 1.3
			5. Completeness	[KF] 7.3, [K] 1.4
5	09/23 - 09/27	48-59	6. Three applications of completeness	[KF] 6.3, [K] 1.5
			7. Completion of a metric space	[KF] 7.4, [K] 1.6
			8. Compactness	[KF] 10.1, 10.2, 10.3, 10.4, [K] 2.5
6	09/30 - 10/04	60-76	III. Linear Spaces
			1. Linear spaces	[KF] 13.1, 13.2, 13.3, [K] 2.1
			2. Frechet spaces	[KF] 17.1, 17.2
			3. Banach spaces	[KF] 15.1, 15.2, [K] 2.2, 2.3
7	10/07 - 10/11	77-95	4. Finite-dimensional Banach spaces	[K] 2.4
			5. Compactness of the unit sphere	[K] 2.5
			6. Hilbert spaces	[KF] 16.1, 16.2, 16.8, 16.9, [K] 3.1, 3.2
8	10/14		No class! (Fall Break)
	10/15 - 10/18	96-101	7. Orthonormal sets	[KF] 16.3, 16.4, 16.5, 16.6, [K] 3.4, 3.5, 3.6
9	10/21 - 10/25	102-106	8. Orthogonal projections	[KF] 16.7, [K] 3.3
	10/23		Midterm Exam! (4:30-5:45pm)
10	10/28 - 11/01	107-120	9. Superposition principle revisited	[K] 3.5
			IV. Linear Functionals
			1. Continuous linear operators	[KF] 18.1, 18.2, 22.1, 22.2, [K] 2.6, 2.7, 2.8, 2.9
11	11/04 - 11/08	121-133	2. The Banach space L(X,Y)	[KF] 22.3, [K] 2.7
			3. Dual spaces	[KF] 19.1, 19.2, [K] 2.10
12	11/11 - 11/15	134-145	4. Convex sets and functionals	[KF] 14.1, 14.2, 14.3, [K] 2.8
			5. The Hahn-Banach theorem	[KF] 14.4, 18.3, [K] 4.2, 4.3
			V. Linear Operators
13	11/18 - 11/22	146-155	1. Inverse operators	[KF] 23.1
			2. Adjoint operators	[KF] 23.2, 23.3
14	11/25 - 11/26	156-	3. Solvability conditions
	11/27		No class! (Thanksgiving)
15	12/02 - 12/06		4. Spectrum and resolvent	[KF] 23.4
			5. Completely continuous operators	[KF] 24.1, 24.2
16	12/09		6. Spectral theory of compact operators	[KF] 24.3
	12/11		Final Exam! (4:30-6:30pm)

The following two books are the main texts used for the course:

E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978. [K]
A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover, 1970. [KF]

While the book [KF] forms the basis for the preliminary exam schedule, it is rather terse and has not been a favorite of recent students. I personally find [K] a very readable and complete text, but it is unfortunately very expensive. In addition, the following list contains a number of books that you might find useful for supplementary reading. All of these texts cover the basic material of the course. Note that the books by MacCluer and by Rynne & Youngson are available for free as pdf downloads via the ebook subscription of the library.

B. Bollobas, Linear Analysis: An Introductory Course, Cambridge University Press, 1999 (2nd edition).
P. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013.
T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.
P.D. Lax, Functional Analysis, Wiley, 2002.
B.D. MacCluer, Elementary Functional Analysis, Springer, 2009.
G.K. Pedersen, Analysis Now, Springer, 1989.
F. Riesz, B. Sz.-Nagy, Functional Analysis, Dover, 1990.
B.P. Rynne, M.A. Youngson, Linear Functional Analysis, Springer, 2008 (2nd edition).
A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, Robert E. Krieger Publishing, 1986 (2nd edition).
K. Yosida, Functional Analysis, Springer, 1980 (6th edition).