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.
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.
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.