MAGIC061: Functional Analysis

Course details

A core MAGIC course

Semester

Autumn 2023
Monday, October 2nd to Friday, December 8th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
80

Timetable

Wednesdays
12:05 - 12:55 (UK)
Wednesdays
13:05 - 13:55 (UK)

Description

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 and the spectral theorm for compact normal 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 anticipates:

  • 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, Engineering and Statistics applications...

Some relevant books. (See the Bibliography page for more details of these books.) 
 
G. K. Pedersen, Analysis Now (Springer, 1988) 
 [This course may be viewed as a preparation for studying this text (which is already a classic).] 

 G.F. Simmons, 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. 

Prerequisites

Standard undergraduate linear algebra and real and complex analysis, and basic metric space/norm topology.

Syllabus

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. 
 Stone-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 operator; 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; Spectral Theorem for compact normal operators. 

 APPENDICES 

 Nets and generalised sums. 

 Topological vector spaces. 

Lecturer

  • ML

    Professor Martin Lindsay

    University
    University of Lancaster

Bibliography

Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

Assessment

The assessment for this course will be released on Monday 8th January 2024 at 00:00 and is due in before Friday 19th January 2024 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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