Robert Gardner's Online Fundamentals of Functional Analysis

Online Fundamentals of Functional Analysis
with online videos Dr. Bob's Online University
(Not an actual university)
Robert 'Dr. Bob' Gardner

This website includes links to class notes and supplements (in PDF) used in the teaching of East Tennessee State University's Fundamentals of Functional Analysis (MATH 5740). Links are also available for video presentations of the notes. The videos have been used as of the teaching of online Fundamentals of Functional Analysis, starting in summer 2021. The material is largely based on S. David Promislow's A First Course In Functional Analysis, Hoboken, NJ: John Wiley & Sons (2008):

Promislow's A First Course In Functional Analysis book

Robert "Dr. Bob" Gardner can be reached by e-mail at: gardnerr@etsu.edu. His office is in Gilbreath Hall, Room 308F on the ETSU campus.

The following notes and supplements are all in PDF. The videos can be streamed from YouTube and should remain permanently available. (The videos previously on ETSU's Panopto host site have been, or may soon be, deleted.)

1. Linear Spaces and Operators.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
1.1. Introduction	1.1 Notes	-	Section 1.1 Video (14:05)
1.2. Linear Spaces	1.2 Notes	-	Section 1.2 Video (30:59)
1.3. Linear Operators	1.3 Notes	1.3 Supplement 1.3 Supplement Print File	Section 1.3 Video (36:17)
1.4. Passage from Finite- to Infinite-Dimensional Spaces	1.4 Notes	-	Section 1.4 Video (4:51)

2. Normed Linear Spaces: The Basics.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
2.1. Metric Spaces	2.1 Notes	-	Section 2.1 Video (17:36)
2.2. Norms	2.2 Notes	2.2 Supplement 2.2 Supplement Print File	Section 2.2 Video (1:19:35)
2.3. Space of Bounded Functions	2.3 Notes	2.3 Supplement 2.3 Supplement Print File	Section 2.3 Video (13:50)
2.4. Bounded Linear Operators	2.4 Notes	2.4 Supplement 2.4 Supplement Print File	Section 2.4 Video (49:43)
2.5. Completeness	2.5 Notes	2.5 Supplement 2.5 Supplement Print File	Section 2.5 Video, Part 1 (58:00) Section 2.5 Video, Part 2 (24:47) Section 2.5 Video, Part 3 (46:42)*
2.6. Comparison of Norms	2.6 Notes	2.6 Supplement 2.6 Supplement Print File	Section 2.6 Video (29:23)
2.7. Quotient Spaces	2.7 Notes	2.7 Supplement 2.7 Supplement Print File	Section 2.7 Video (1:15:26)
2.8. Finite-Dimensional Normed Linear Spaces	2.8 Notes	2.8 Supplement 2.8 Supplement Print File	Section 2.8 Video, Part 1 (1:03:09) Section 2.8 Video, Part 2 (32:07)
2.9. L^p Spaces	2.9 Notes	-	Section 2.9 Video (1:08:29)
2.10. Direct Sums and Products	2.10 Notes	-	Section 2.10 Video (13:33)
2.11. Schauder Bases	2.11 Notes	-	Section 2.11 Video (9:17)
Supplement. Groups, Fields and Vector Spaces	Supplement Notes	Supplement Supplement Print File	Groups, etc. Video, Part 1 (1:01:40) Groups, etc. Video, Part 2 (41:50)
2.12. Fixed Points and Contraction Mappings	2.12 Notes	2.12 Supplement 2.12 Supplement Print File	Section 2.12 Video (16:01)

*The proof of Theorem 2.22 only addresses uniqueness. See the revised online notes for details.

3. Major Banach Space Theorems.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
3.1. Introduction	3.1 Notes	-	Section 3.1 Video (1:58)
3.2. Baire Category Theorem	3.2 Notes	3.2 Supplement 3.2 Supplement Print File	Section 3.2 Video (29:20)
3.3. Open Mappings	3.3 Notes	3.3 Supplement 3.3 Supplement Print File	Section 3.3 Video (47:02)
3.4. Bounded Inverses	3.4 Notes	3.4 Supplement 3.4 Supplement Print File	Section 3.4 Video (20:56)
3.5. Closed Linear Operators	3.5 Notes	3.5 Supplement 3.5 Supplement Print File	Section 3.5 Video (32:32)
3.6. Uniform Boundedness Principle	3.6 Notes	3.6 Supplement 3.6 Supplement Print File	Section 3.6 Video (37:14)

4. Hilbert Spaces.

SECTION	NOTES	SUPPLEMENTS	VIDEOS ON YOUTUBE
4.1. Introduction	4.1 Notes	-	Section 4.1 Video (5:34)
4.2. Semi-Inner Products	4.2 Notes	4.2 Supplement 4.2 Supplement Print File	Section 4.2 Video (1:02:48)
Supplement. Inner Products	Supplement Notes	Supplement Supplement Print File	Inner Products Video (56:30)
4.3. Nearest Points and Convexity	4.3 Notes	4.3 Supplement 4.3 Supplement Print File	Section 4.3 Video (43:49)
4.4. Orthogonality	4.4 Notes	4.4 Supplement 4.4 Supplement Print File	Section 4.4 Video (1:10:34)*
Supplement. The Space L^p	Supplement Notes	-	L^p Video (10:41)
Supplement. Projections and Hilbert Space Isomorphisms	Supplement Notes	Supplement Supplement Print File	Projections Video, Part 1 (1:03:52) Projections Video, Part 2 (46:20)
4.5. Linear Functionals on Hilbert Space	4.5 Notes	-	Section 4.5 Video (14:44)
Supplement. A Proof of Theorem 4.22	Supplement Notes	-	-
4.6. Linear Operators in Hilbert Spaces	4.6 Notes	4.6 Supplement 4.6 Supplement Print File	-
4.7. Order Relation on Self-Adjoint Operators	4.7 Notes	4.7 Supplement 4.7 Supplement Print File	-

*The definition of "orthonormal basis" is modified in the notes (the condition "countable" should not have been present).

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Last updated: March 23, 2024.