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 the prerecorded lectures on my YouTube channel.
Key organizational information:
Instructor: Thomas Bothner
Office: 2A.15, Fry Building (not during COVID-19 lockdowns)
Contact: thomas.bothner@bristol.ac.uk and 0117 428 4992
This course will be taught in a blended fashion: all lectures are pre-recorded, see below, and we will have one live zoom problem class each Friday afternoon. In addition, Alex Little will run an online Maths Cafe session every Thursday evening and I will have online office hours every Tuesday. Further details, including learning outcomes and assessment criteria, are summarized in the syllabus. Please note, all zoom meeting links will be announced on Blackboard.
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 2021 written exam, see our Blackboard page for previous exams.
Course materials (click the links):
Welcome to Functional Analysis - introductory slides
Lecture Notes (February 19th, 2021 version - let me know about typos)
Exercises (solutions on Blackboard)
Weekly schedule: (Office Hour=OH, Problem Class=PC, Maths Cafe=MC)
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Week 13, February 1st - February 5th:
Topics: Normed linear spaces, bounded linear transformations, equivalent norms, Hoelder and Minkowski inequalities, operator norms, examples (page 8-17 in the lecture notes)
Watch: w13v1, w13v2, w13v3, w13v4, w13v5, w13v6
Do: Exercise sheet 1, to be discussed in PC on Feb. 12th
Meet: OH, Feb. 2nd (16:00-17:00); PC, Feb. 5th (14:00-15:00)
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Week 14, February 8th - February 12th:
Topics: Integral operators, 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)
Watch: w14v1, w14v2, w14v3, w14v4, w14v5, w14v6
Do: Exercise sheet 1, to be discussed in PC on Feb. 12th
Meet: OH, Feb. 9th (16:00-17:00); MC, Feb. 11th (17:00-18:00); PC, Feb. 12th (14:00-15:00)
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Week 15, February 15th - February 19th:
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)
Watch: w15v1, w15v2, w15v3, w15v4, w15v5, w15v6
Do: Exercise sheet 2, to be discussed in PC on Feb. 19th
Meet: OH, Feb. 16th (16:00-17:00); MC, Feb. 18th (17:00-18:00); PC, Feb. 19th (14:00-15:00)
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Week 16, February 22nd - February 26th:
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)
Watch: w16v1, w16v2, w16v3, w16v4, w16v5, w16v6
Do: Exercise sheet 3, to be discussed in PC on Feb. 26th
Meet: OH, Feb. 23rd (16:00-17:00); MC, Feb. 25th (17:00-18:00); PC, Feb. 26th (14:00-15:00)
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Week 17, March 1st - March 5th:
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)
Watch: w17v1, w17v2, w17v3, w17v4, w17v5, w17v6
Do: Assessed HW 1 (due March 5th, noon) and exercise sheet 4, to be discussed in PC on Mar. 5th
Meet: OH, Mar. 2nd (16:00-17:00); MC, Mar. 4th (17:00-18:00); PC, Mar. 5th (14:00-15:00)
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Week 18, March 8th - March 12th:
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)
Watch: w18v1, w18v2, w18v3, w18v4, w18v5, w18v6
Do: Exercise sheet 5, to be discussed in PC on Mar. 12th
Meet: OH, Mar. 9th (16:00-17:00); MC, Mar. 11th (17:00-18:00); PC, Mar. 12th (14:00-15:00)
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Week 19, March 15th - March 19th:
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)
Watch: w19v1, w19v2, w19v3, w19v4, w19v5, w19v6
Do: Exercise sheet 6, to be discussed in PC on Mar. 19th
Meet: OH, Mar. 16th (16:00-17:00); MC, Mar. 18th (17:00-18:00); PC, Mar. 19th (14:00-15:00)
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Week 20, March 22nd - March 26th:
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)
Watch: w20v1, w20v2, w20v3, w20v4
Do: Exercise sheet 7, to be discussed in PC on Mar. 26th
Meet: OH, Mar. 23rd (16:00-17:00); MC, Mar. 25th (17:00-18:00); PC, Mar. 26th (14:00-15:00)
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Week 21, April 19th - April 23rd:
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)
Watch: w21v1, w21v2, w21v3, w21v4
Do: Exercise sheet 8, to be discussed in PC on Apr. 23rd
Meet: OH, Apr. 20th (16:00-17:00); MC, Apr. 22nd (17:00-18:00); PC, Apr. 23rd (14:00-15:00)
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Week 22, April 26th - April 30th:
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)
Watch: w22v1, w22v2, w22v3, w22v4
Do: Assessed HW 2 (due April 30th, noon) and exercise sheet 9, to be discussed in PC on Apr. 30th
Meet: OH, Apr. 27th (16:00-17:00); MC, Apr. 29th (17:00-18:00); PC, Apr. 30th (14:00-15:00)
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Week 23, May 3rd - May 7th:
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 10, to be discussed in PC on May 7th
Meet: OH, May 4th (16:00-17:00); MC, May 6th (17:00-18:00); PC, May 7th (14:00-15:00)
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Week 24, May 10th - May 14th:
Topics: Riesz-Schauder theorem, Hilbert-Schmidt theorem, spectral decomposition theorem (page 83-87 in the lecture notes)
Watch: w24v1, w24v2, w24v3, w24v4
Do: Exercise sheet 11, to be discussed in PC on May 14th
Meet: OH, May 11th (16:00-17:00); MC, May 13th (17:00-18:00); PC, May 14th (14:00-15:00)