The goal of this lecture is to understand the following kind of questions:
- Let X be a (second countable) compact topological space, and T be a homeomorphism from X to itself; why does X admit a T-invariant Borel probability measure?
- Let V(t) be a continuous real function for 0 ≤ t ≤ 1, and consider the differential equation \(- \frac{d^2 u}{d t^2}(t) + V(t) u(t) = \alpha u(t)\) for (nonzero) complex valued function u(t) satisfying u(0) = u(1) = 0. Why should α belong to a discrete subset of R?
More formally, the first part of this course is about locally convex spaces (mostly Banach spaces and their dual spaces). In the second half we cover the basic theory on operators on Hilbert spaces, such as sconcrete and abstract pectral theory (theory of eigenvalues), and unbounded operators. These topics grew out of several important fields of classical analysis such as Dirichlet problem (2. above), but these will also be essential for more modern and abstract fields such as the study of operator algebras.
Requirements
You need to be already familiar with the content of MAT4410.
References
Any book on functional analysis would do, but a concise reference is
- Ronald G. Douglas, Banach algebra techniques in operator theory, second ed., Graduate Texts in Mathematics, vol. 179, Springer-Verlag, New York, 1998. (Chapters 1–2)
More comprehensive ones include:
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John B. Conway, A course in functional analysis, second ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990.
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Nicolas Bourbaki, ?léments de mathématique. Espaces vectoriels topologiques. & Theories spectrales., Springer, 2007
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Gert K. Pedersen, Analysis now, Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989.
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Michael Reed and Barry Simon, Methods of modern mathematical physics. Academic Press, New York-London, 1972, 1975.
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Walter Rudin, Functional analysis, second ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.