Functional Analysis 

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:

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):

Weekly schedule: (Office Hour=OH, Class=C)

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Week 13, January 22nd - January 26th: 

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. 22nd, 24th, 25th

Meet: OH, Jan. 25th (13:00-14:00); C, Jan. 22nd (16:00-17:00), Jan. 24th (10:00-11:00), Jan. 25th (11:00-12:00)

Can watch as support: w13v1, w13v2, w13v3, w13v4, w13v5, w13v6

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Week 14, January 29th - February 2nd: 

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. 29th, 31st, Feb. 1st

Meet: OH, Feb. 1st (13:00-14:00); C, Jan. 29th (16:00-17:00), Jan. 31st (10:00-11:00), Feb. 1st (11:00-12:00)

Can watch as support: w14v1, w14v2, w14v3, w14v4, w14v5, w14v6

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Week 15, February 5th - February 9th:

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: Assessed HW1 (due Feb. 9th, noon), Exercise sheet 2 and 3, to be worked on in C on Feb. 5th, 7th, 8th

Meet: OH, Feb. 8th (13:00-14:00); C, Feb. 5th (16:00-17:00), Feb. 7th (10:00-11:00), Feb. 8th (11:00-12:00)

Can watch as support: w15v1, w15v2, w15v3, w15v4, w15v5, w15v6

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Week 16, February 12th - February 16th:

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. 12th, 14th, 15th

Meet: OH, Feb. 15th (13:00-14:00); C, Feb. 12th (16:00-17:00), Feb. 14th (10:00-11:00), Feb. 15th (11:00-12:00)

Can watch as support: w16v1, w16v2, w16v3, w16v4, w16v5, w16v6

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Week 17, February 19th - February 23rd:

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: Exercise sheet 5, to be worked on in C on Feb. 19th, 21st, 22nd

Meet: OH, Feb. 22nd (13:00-14:00); C, Feb. 19th (16:00-17:00), Feb. 21st (10:00-11:00), Feb. 22nd (11:00-12:00)

Can watch as support: w17v1, w17v2, w17v3, w17v4, w17v5, w17v6

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Week 18, February 26th - March 1st:

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 (page 47-54 in the lecture notes)

Do: Assessed HW 2 (due Mar. 1st, noon), Exercise sheet 6, to be worked on in C on Feb. 26th, 28th, 29th

Meet: OH, Mar. 1st (13:00-14:00); C, Feb. 26th (16:00-17:00), Feb. 28th (10:00-11:00), Feb. 29th (11:00-12:00)

Can Watch as support: w18v1, w18v2, w18v3, w18v4, w18v5, w18v6

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Week 19, March 4th - March 8th:

Topics: Graphs of linear maps, closed graph theorem, Hellinger-Toeplitz theorem, weak convergence, weak* convergence, properties, Helly's theorem, topologies on bounded operators, Hilbert space adjoint (page 55-62 in the lecture notes)

Do: Exercise sheet 7, to be worked on in C on Mar. 4th, 6th, 7th

Meet: OH, Mar. 7th (13:00-14:00); C, Mar. 4th (16:00-17:00), Mar. 6th (10:00-11:00), Mar. 7th (11:00-12:00)

Can watch as support: w19v1, w19v2, w19v3, w19v4, w19v5, w19v6

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Week 20, March 11th - March 15th:

Topics: Properties of the Hilbert space adjoint, examples, normal operators, self-adjoint operators, unitary transformations, criteria for self-adjoint operators and unitary transformations, projections (page 63-68 in the lecture notes)

Do: Exercise sheet 8, to be worked on in C on Mar. 11th, 13th, 14th

Meet: OH, Mar. 14th (13:00-14:00); C, Mar. 11th (16:00-17:00), Mar. 13th (10:00-11:00), Mar. 15th (11:00-12:00)

Can watch as support: w20v1, w20v2, w20v3, w20v4

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Week 21, March 18th - March 22nd:

Topics: Resolvent set, resolvent operator, spectrum, continuous spectrum, residual spectrum, examples, resolvent formula, non-emptiness of spectrum, spectral radius formula (page 68-72 in the lecture notes)

Do: Assessed HW 3 (due Mar. 22nd, noon), Exercise sheet 9, to be worked on in C on Mar. 18th, 20th, 21st

Meet: OH, Mar. 21st (13:00-14:00); C, Mar. 18th (16:00-17:00), Mar. 20th (10:00-11:00), Mar. 21st (11:00-12:00)

Can watch as support: w21v1, w21v2, w21v3, w21v4

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Week 22, April 15th - April 19th:

Topics: Example 31, spectral properties of normal , unitary and self-adjoint maps on Hilbert spaces, compact linear transformations, finite rank transformations, properties, Schauder's theorem, example (page 73-77 in the lecture notes)

Do: Exercise sheet 10, to be worked on in C on Apr. 15th, 17th, 18th

Meet: OH, Apr. 18th (13:00-14:00); C, Apr. 15th (16:00-17:00), Apr. 17th (10:00-11:00), Apr. 18th (11:00-12:00)

Can watch as support: w22v1, w22v2, w22v3, w22v4

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Week 23, April 22nd- April 26th:

Topics: Theorem 45, Riesz geometric lemma, finite approximable linear transformations, properties, Approximation property, Fredholm Alternative (page 78-82 in the lecture notes)

Watch: w23v1, w23v2, w23v3, w23v4

Do: Exercise sheet 11, to be worked on C on Apr. 22nd, 24th, 25th

Meet: OH, Apr. 25th (13:00-14:00); C, Apr. 22nd (16:00-17:00), Apr. 24th (10-:00-11:00), Apr. 25th (11:00-12:00)

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Week 24, April 29th - May 3rd:

Topics: Riesz-Schauder theorem, Hilbert-Schmidt theorem, spectral decomposition theorem  (page 83-87 in the lecture notes)

Watch: w24v1, w24v2, w24v3, w24v4

Do: Assessed HW 4 (due May 3rd, noon)

Meet: OH, May 2nd (13:00-14:00); C, Apr. 29th (16:00-17:00), May 1st (10:00-11:00), May 2nd (11:00-12:00)