Pure Functional Analysis (MAGIC061)
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This course is part of the MAGIC core.
This couse provides an introduction to analysis in infinite dimensions with a minimum of prerequisites. The core of the course concerns operators on a Hilbert space including the continuous functional calculus for bounded selfadjoint operators. There will be an emphasis on positivity and on matrices of operators. The course includes some basic introductory material on Banach spaces and Banach algebras. It also includes some elementary (infinite dimensional) linear algebra that is usually excluded from undergraduate curricula. Here is a very brief list of the many further topics that this course looks forward to. Banach space theory and Banach algebras; C*-algebras, von Neumann algebras and operator spaces (which may be viewed respectively as noncommutative topology, noncommutative measure theory and `quantised' functional analysis); Hilbert C*-modules; noncommutative probability (e.g. free probability), the theory of quantum computing, dilation theory; unbounded Hilbert space operators, one-parameter semigroups and Schrodinger operators. And that is without starting to mention Applied Maths and Statistics applications ... Some relevant books. (See the Bibliography page for more details of these books.) G. K. Pederson, Analysis Now (Springer, 1988) [This course may be viewed as a preparation for studying this text (which is already a classic).] Simmonds, Introduction to Topology and Modern Analysis (McGraw-Hill, 1963) [Covers far more than the course, but is still distinguished by its great accessibility.] P.R. Halmos, Hilbert Space Problem Book (Springer, 1982) [Collected and developed by a master expositor.] There are many many other books which cover the core part of this course.
Autumn 2011 (Monday, October 10 to Friday, December 16)
Standard undergraduate linear algebra and real and complex analysis, and basic metric space/norm topology.
I PRELIMINARIES Linear Algebra. Including quotient space and free vector space constructions, diagonalisation of hermitian matrices, algebras, homomorphisms and ideals, group of units and spectrum. Metric Space. Review of basic properties, including completeness and extension of uniformly continuous functions. General Topology. Including compactness and Polish spaces. Banach Space. Including dual spaces, bounded operators, bidual [and weak*-topology], completion and continuous (linear) extension. Banach Algebra. Including Neumann series, continuity of inversion, spectrum, C*-algebra definition. Hilbert Space Geometry. Including Bessel's inequality, dimension, orthogonal complementation, nearest point projection for nonempty closed convex sets. Miscellaneous. Including Weierstrass Approximation Theorem. II HILBERT SPACE AND ITS OPERATORS Sesquilinearity, orthogonal projection; Riesz-Frechet Theorem, adjoint operators, C*-property; Kernel-adjoint-range relation; Finite rank operators; Operator types: normal, unitary, selfadjoint, isometric, compact, invertible, nonnegative, uniformly positive and partially isometric; Fourier transform as unitary operator; Invertibility criteria; Key examples of operators, finding their spectra (shifts and multiplication operators), norm and spectrum for a selfadjoint; III FURTHER TOPICS AS TIME PERMITS. Polar decomposition. Continuous functional calculus for selfadjoint operators, with key examples: square-root and positive/negative parts. Matrices of operators, positivity in B(h+k), operator space - definition and simple examples. Nonnegative-definite kernels, Kolmogorov decomposition; Hilbert space tensor products; Hilbert-Schmidt operators. Topologies on spaces of operators (WOT, SOT, uw). Compact and trace class operators, duality. Double Commutant Theorem. Dilation and von Neumann's inequality. Two projections in general position.
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*** UPDATED on 15 December 2011 *** The Lecture Notes are laced with numerous Exercises. You are asked to do as many of these as you can - starting with the most challenging ones. As a very rough guide, you could aim to submit solutions to the two toughest questions that you can manage from each chapter/topic (Chapters 1 - 8). Some partial answers to the more challenging questions will be acceptable. If you wish to submit solutions to exercises from the extra material in Chapters 9 or 10, then that is fine - these can be in place of exercises from earlier material. Please submit your work in electronic form as a pdf file, preferably produced from a TeX source file. *** Liberalised Deadline: 31 January 2012 ***
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