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 2026 written exam, see our Blackboard page for previous exams.
Course materials (click the links):
Welcome to Functional Analysis - Introductory slides
Lecture Notes (January 23rd, 2026 version - let me know about typos)
Exercises (solutions on Blackboard, once discussed in class)
Weekly schedule: (Office Hour=OH, Class=C, Support Session=SP)
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Week 13, January 19th - January 23rd:
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 19th, 20th, 23rd
Meet: OH, Jan 19th (11:30-12:30); C, Jan 19th (10:00-11:00), Jan 22nd (15:00-16:00), Jan 23rd (11:00-12:00); SP, Jan 20th (14:00-15:00)
Can watch as support: w13v1, w13v2, w13v3, w13v4, w13v5, w13v6
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Week 14, January 26th - January 30th:
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 (page 17-22 in the lecture notes)
Do: Exercise sheet 1, to be worked on in C on Jan 26th, 29th, 30th
Meet: OH, Jan 26th (11:30-12:30); C, Jan 26th (10:00-11:00), Jan 29th (15:00-16:00), Jan 30th (11:00-12:00); SP, Jan 27th (14:00-15:00)
Can watch as support: w14v1, w14v2, w14v3, w14v4, w14v5, w14v6
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Week 15, February 2nd - February 6th:
Topics: Best approximation, orthogonal complements, projection lemma, Riesz representation, orthonormal basis, Fourier expansion theorem, separable Hilbert spaces, linearly independent spanning sets, Gram-Schmidt procedure, isomorphic Hilbert spaces (page 23-29 in the lecture notes)
Do: Exercise sheet 2;, to be worked on in C on Feb 2nd, 5th, 6th
Meet: OH, Feb 2nd (11:30-12:30); C, Feb 2nd (10:00-11:00), Feb 5th (15:00-16:00), Feb 6th (11:00-12:00); SP, Feb 3rd (14:00-15:00)
Can watch as support: w15v1, w15v2, w15v3, w15v4, w15v5, w15v6
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Week 16, February 9th - February 13th:
Topics: Banach spaces, summable sequences, invertible transformations, examples, Neumann series, densely defined linear transformations and the BLT theorem, Hahn-Banach theorems, corollaries (3.1-3.3) to Hahn-Banach (page 34-40 in the lecture notes)
Do: Exercise sheet 3 and 4, to be worked on in C on Feb 9th, 12th, 13th
Meet: OH, Feb 9th (11:30-12:30); C, Feb 9th (10:00-11:00), Feb 12th (15:00-16:00), Feb 13th (11:00-12:00); SP, Feb 10th (14:00-15:00)
Can watch as support: w16v1, w16v2, w16v3, w16v4, w16v5, w16v6
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Week 17, February 16th - February 20th:
Topics: Corollaries to Hahn-Banach (3.4,3.5 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 19th, noon); Exercise sheet 5, to be worked on in C on Feb 16th, 19th, 20th
Meet: OH, Feb 16th (11:30-12:30); C, Feb 16th (10:00-11:00), Feb 19th (15:00-16:00), Feb 20th (11:00-12:00); SP, Feb 17th (14:00-15:00)
Can watch as support: w17v1, w17v2, w17v3, w17v4, w17v5, w17v6