|Unit level:||Level 3|
|Teaching period(s):||Semester 1|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
- MATH20101 - Real and Complex Analysis (Compulsory)
- MATH20111 - Real Analysis (Compulsory)
- MATH20122 - Metric Spaces (Compulsory)
Students must have taken MATH20122 and (MATH20101 OR MATH20111)
To give an introduction to Modern Analysis, including elements of functional analysis. The emphasis will be on ideas and results.
This course unit deals with a coherent and elegant collection of results in analysis. The aim of this course is to provide an introduction to the theory of infinite dimensional linear spaces, which is not only an important tool, but is also a central topic in modern mathematics.
This area has many applications to other areas in Pure and Applied Mathematics such as Dynamical Systems, C* algebras, Quantum Physics, Numerical Analysis, etc.
On successful completion of this module students will be able to
- understand the approximation of continuous functions using the Stone-Weierstrass Theorem;
- understand the concepts of Hilbert and Banach spaces, with l 2 and l p spaces serving as examples;
- understand the definitions of linear functionals and dual spaces, prove the Riesz representation theorem for Hilbert spaces;
- define linear operators, their spectrum (with matrices serving as examples) and their spectral radius, understand the definitions of self-adjoint, isometric and unitary operators on Hilbert spaces and their spectra, be able to apply these ideas to matrices.
Students will have seen proofs of the main results, although an intimate knowledge of the full details is not required.
Future topics requiring this course unit
This course would be useful to students interested in the following topics: MATH41012 Fourier Analysis and Lebesgue Integration.
Assessment Further Information
End of semester examination: two hours weighting 100%
1.Revision of compactness and uniform continuity. [2 lectures]
2.Approximation by polynomials and the Stone-Weierstrass Theorem. 
3.Normed vector spaces. Finite and infinite dimensional spaces. Completeness and Banach spaces. 
4.Sequence spaces. Continuous function spaces. Separability. Hilbert spaces, orthogonal complements and direct sum decompositions, orthonormal bases. 
5.Bounded linear functionals. The Hahn-Banach theorem. Dual spaces. The Riesz representation theorem for Hilbert spaces. 
6.Bounded linear operators and their norms. Compact operators. Invertible operators. Spectra. 
7.Linear operators on Hilbert spaces. Self-adjoint and unitary operators and their spectra. 
- G. F. Simmons, An Introduction to Topology and Modern Analysis, McGraw Hill, 1963.
- I.J. Maddox, Elements of Functional Analysis, C.U.P. 1989.
- B.P. Rynne and M.A. Youngson, Linear Functional Analysis, Springer S.U.M.S., 2008.
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours