Welcome to Functional Analysis, MATH36202 and MATHM6202, a double batched module for undergraduate students in year 3 and 4 at the University of Bristol. This page summarizes all key information and contains most course materials, most importantly our weekly schedule with links to recorded lectures on my YouTube channel.
Key organizational information:
Instructor: Thomas Bothner
Office: 2A.15, Fry Building
Contact: thomas.bothner@bristol.ac.uk and +44 117 455 2070
This course will be taught in a semi-traditional fashion: you will acquire new theoretical knowledge in the classroom but we will focus foremost on concepts and practical work. Content reaches you through lecture notes, in-class lesson notes, Blackboard learning materials and a series of supporting videos. Further details, including learning outcomes and assessment criteria, are summarized in the syllabus.
Depending on whether you are enrolled in MATH36202 or MATHM6202, please familiarize yourself with the unit descriptions, click here for MATH36202 and here for MATHM6202. Throughout, there will be no distinction between the two levels as far as course content, exercises or assessed homeworks are concerned. However, there will be a slight difference between MATH36202 and MATHM6202 in the May/June 2024 written exam, see our Blackboard page for previous exams.
Course materials (click the links):
Welcome to Functional Analysis - Introductory slides
Lecture Notes (January 16th, 2025 version - let me know about typos)
Exercises (solutions on Blackboard, once discussed in class)
Weekly schedule: (Office Hour=OH, Class=C, Support Session=SP)
------------------------------------------------------------------------------------------------------------------
Week 13, January 13th - January 17th:
Topics: Normed linear spaces, bounded linear transformations, equivalent norms, Hoelder and Minkowski inequalities, operator norms, examples (page 8-17 in the lecture notes)
Do: Exercise sheet 1, to be worked on in C on Jan 13th, 14th, 15th
Meet: OH, Jan 14th (11:30-12:30); C, Jan 13th (12:00-13:00), Jan 14th (13:00-14:00), Jan 15th (12:00-13:00)
Can watch as support: w13v1, w13v2, w13v3, w13v4, w13v5, w13v6
------------------------------------------------------------------------------------------------------------------
Week 14, January 20th - January 24th:
Topics: Inner product spaces, examples, orthogonality and orthonormal sets, parallelogram identity, Pythagorean theorem, Bessel's inequality, Cauchy-Schwarz inequality, Jordan-von Neumann theorem, Hilbert spaces, examples, best approximation, orthogonal complements, projection lemma, Riesz representation (page 17-25 in the lecture notes)
Do: Exercise sheet 1, to be worked on in C on Jan 24th, 25th, 26th
Meet: OH, Jan 25th (11:30-12:30); C, Jan 24th (12:00-13:00), Jan 25th (13:00-14:00), Jan 26th (12:00-13:00); SP, Jan 25th (09:00-10:00)
Can watch as support: w14v1, w14v2, w14v3, w14v4, w14v5, w14v6
------------------------------------------------------------------------------------------------------------------
Week 15, January 27th - January 31st:
Topics: Zorn's lemma, orthonormal basis, Fourier expansion theorem, separable Hilbert spaces, linearly independent spanning sets, Gram-Schmidt procedure, isomorphic Hilbert spaces, examples (page 26-33 in the lecture notes)
Do: Exercise sheet 2 and 3, to be worked on in C on Jan 27th, 28th, 29th
Meet: OH, Jan 28th (11:30-12:30); C, Jan 27th (12:00-13:00), Jan 28th (13:00-14:00), Jan 29th (12:00-13:00); SP, Jan 27th (09:00-10:00)
Can watch as support: w15v1, w15v2, w15v3, w15v4, w15v5, w15v6
----------------------------------------------------------------------------------------------------------------
Week 16, February 3rd - February 7th:
Topics: Summable sequences, invertible transformations, examples, Neumann series, densely defined linear transformations and the BLT theorem, Hahn-Banach theorems, corollaries (7-9) to Hahn-Banach (page 34-40 in the lecture notes)
Do: Exercise sheet 4, to be worked on in C on Feb 3rd, 4th, 5th
Meet: OH, Feb 4th (11:30-12:30); C, Feb 3rd (12:00-13:00), Feb 4th (13:00-14:00), Feb 5th (12:00-13:00); SP, Feb 4th (09:00-10:00)
Can watch as support: w16v1, w16v2, w16v3, w16v4, w16v5, w16v6
----------------------------------------------------------------------------------------------------------------
Week 17, February 10th - February 14th:
Topics: Corollaries to Hahn-Banach (10,11 and Theorem 19), dual transformation, bidual, canonical map, reflexive Banach spaces, properties, examples (page 41-46 in the lecture notes)
Do: Assessed HW1 (due February 13th, noon); Exercise sheet 5, to be worked on in C on Feb 10th, 11th, 12th
Meet: OH, Feb 11th (11:30-12:30); C, Feb 10th (12:00-13:00), Feb 11th (13:00-14:00), Feb 12th (12:00-13:00); SP, Feb 11th (09:00-10:00)
Can watch as support: w17v1, w17v2, w17v3, w17v4, w17v5, w17v6
----------------------------------------------------------------------------------------------------------------
Week 18, February 17th - February 21st:
Topics: Consolidation week
Do: Exercise sheet 6, to be worked on in C on Feb 17th, 18th, 19th
Meet: OH, Feb 18th (11:30-12:30); C, Feb 17th (12:00-13:00), Feb 18th (13:00-14:00), Feb 19th (12:00-13:00); SP, Feb 18th (09:00-10:00)
---------------------------------------------------------------------------------------------------------------
Week 19, February 24th - February 28th:
Topics: Examples of dual spaces, Cantor's theorem, Baire category theorem, principle of uniform boundedness, open mapping theorem, norm equivalence, direct sums and complementary subspaces, graphs of linear maps, closed graph theorem, Hellinger-Toeplitz theorem, weak convergence, properties (page 47-57 in the lecture notes)
Do: Exercise sheet 7, to be worked on in C on Feb 24th, 25th, 26th
Meet: OH, Feb 25th (11:30-12:30); C, Feb 24th (12:00-13:00), Feb 25th (13:00-14:00), Feb 26th (12:00-13:00); SP, Feb 25th (09:00-10:00)
Can Watch as support: w18v1, w18v2, w18v3, w18v4, w18v5, w18v6
---------------------------------------------------------------------------------------------------------------
Week 20, March 3rd - March 7th:
Topics: Weak* convergence, properties, Helly's theorem, topologies on bounded operators, Hilbert space adjoint, properties of the Hilbert space adjoint, examples, normal operators, self-adjoint operators, unitary transformations, criteria for self-adjoint operators and unitary transformations, projections (page 57-67 in the lecture notes)
Do: Exercise sheet 8, to be worked on in C on Mar 3rd, 4th, 5th
Meet: OH, Mar 4th (11:30-12:30); C, Mar 3rd (12:00-13:00), Mar 4th (13:00-14:00), Mar 5th (12:00-13:00); SP, Mar 4th (09:00-10:00)
Can watch as support: w19v1, w19v2, w19v3, w19v4, w19v5, w19v6, w20v1, w20v2, w20v3
---------------------------------------------------------------------------------------------------------------
Week 21, March 10th - March 14th:
Topics: Resolvent set, resolvent operator, spectrum, continuous spectrum, residual spectrum, examples, resolvent formula, non-emptiness of spectrum, spectral radius formula (page 68-74 in the lecture notes)
Do: Exercise sheet 9, to be worked on in C on Mar 10th, 11th, 12th
Meet: OH, Mar 11th (11:30-12:30); C, Mar 10th (12:00-13:00), Mar 11th (12:00-13:00), Mar 12th (12:00-13:00); SP, Mar 11th (09:00-10:00)
Can watch as support: w20v4, w21v1, w21v2, w21v3, w21v4
---------------------------------------------------------------------------------------------------------------
Week 22, March 17th - March 21st:
Topics: Compact linear transformations, finite rank transformations, properties, Schauder's theorem, examples, Theorem 45, Riesz geometric lemma (page 75-79 in the lecture notes)
Do: Exercise sheet 10, to be worked on in C on Mar 17th, 18th, 19th
Meet: OH, Mar 18th (11:30-12:30); C, Mar 17th (12:00-13:00), Mar 18th (13:00-14:00), Mar 19th (12:00-13:00); SP, Mar 18th (09:00-10:00)
Can watch as support: w22v1, w22v2, w22v3, w22v4, w23v1
---------------------------------------------------------------------------------------------------------------
Week 23, March 24th- March 28th:
Topics: Finite approximable linear transformations, properties, Approximation property, Fredholm Alternative, Riesz-Schauder theorem, Hilbert-Schmidt theorem, spectral decomposition theorem (page 79-87 in the lecture notes)
Do: Assessed HW2 (due March 27th, noon); Exercise sheet 11, to be worked on C on Mar 24th, 25th, 26th
Meet: OH, Mar 25th (11:30-12:30); C, Mar 24th (12:00-13:00), Mar 25th (13:00-14:00), Mar 26th (12:00-13:00); SP, Mar 25th (09:00-10:00)
Can watch as support: w23v2, w23v3, w23v4, w24v1, w24v2, w24v3, w24v4