Course Materials
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- Lecturer: Profile: Stuart White
Course information
General prerequisites:
Part A Integration is essential; the only concepts which will be used are the convergence theorems and the theorems of Fubini and Tonelli, and the notions of measurable functions, integrable functions, null sets and \(L^p\) spaces. No knowledge is needed of outer measure, or of any particular construction of the integral, or of any proofs. A good working knowledge of Part A Core Analysis (both metric spaces and complex analysis) is expected.
Course term: Michaelmas
Course lecture information: 16 lectures
Course weight: 1
Course level: H
Assessment type: Written Examination
Course overview:
The course provides an introduction to the methods of functional analysis.
It builds on core material in analysis and linear algebra studied in Part A. The focus is on normed spaces and Banach spaces; a brief introduction to Hilbert spaces is included, but a systematic study of such spaces and their special features is deferred to B4.2 Functional Analysis II. The techniques and examples studied in the Part B courses Functional Analysis I and II support, in a variety of ways, many advanced courses, in particular in analysis and partial differential equations, as well as having applications in mathematical physics and other areas.
It builds on core material in analysis and linear algebra studied in Part A. The focus is on normed spaces and Banach spaces; a brief introduction to Hilbert spaces is included, but a systematic study of such spaces and their special features is deferred to B4.2 Functional Analysis II. The techniques and examples studied in the Part B courses Functional Analysis I and II support, in a variety of ways, many advanced courses, in particular in analysis and partial differential equations, as well as having applications in mathematical physics and other areas.
Learning outcomes:
Students will have a firm knowledge of real and complex normed vector spaces, with their geometric and topological properties. They will be familiar with the notions of completeness, separability and density, will know the properties of a Banach space and important examples, and will be able to prove results relating to the Hahn-Banach Theorem. They will have developed an understanding of the theory of bounded linear operators on a Banach space.
Course synopsis:
Brief recall of material from Part A Metric Spaces and Part A Linear Algebra on real and complex normed vector spaces, their geometry and topology and simple examples of completeness. The norm associated with an inner product and its properties. Banach spaces, exemplified by \(\ell^p, L^p, C(K)\), spaces of differentiable functions. Finite-dimensional normed spaces, including equivalence of norms and completeness. Hilbert spaces as a class of Banach spaces having special properties (illustrations, but no proofs); examples (Euclidean spaces, \(\ell^2, L^2\)), projection theorem, Riesz Representation Theorem.
Density. Approximation of functions, Stone-Weierstrass Theorem. Separable spaces; separability of subspaces.
Bounded linear operators, examples (including integral operators). Continuous linear functionals. Dual spaces. Statement of the Hahn-Banach Theorem; applications, including density of subspaces and embedding of a normed space into its second dual. Adjoint operators.
Density. Approximation of functions, Stone-Weierstrass Theorem. Separable spaces; separability of subspaces.
Bounded linear operators, examples (including integral operators). Continuous linear functionals. Dual spaces. Statement of the Hahn-Banach Theorem; applications, including density of subspaces and embedding of a normed space into its second dual. Adjoint operators.
Section outline
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These are the notes from last year. I will be making some relatively small changes, the first iteration of which will be available in week 0.
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This problem sheet serves as preparation for the course and can be solved before the start of the lectures. This sheet is not for submission but solutions are published here.
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This problem sheet relates to the contents of Section 1.1 on Banach spaces in the lecture notes from 2024-25.
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The topic of this sheet is Hilbert spaces and bounded linear operators and corresponds to the second part of chapter 1 and chapter 2 in the 24-25 lecture notes.
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We have used/stated some of these results in the lectures. It's expected that you familiarise yourself with the statement of these results and will be able to use them comfortably. The proofs themselves are off-syllabus.
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