|Credits: 10||Convenor: Dr. J. Watters||Semester: 2|
|Prerequisites:||essential: MC240, MC241|
|Assessment:||Coursework: 20%||One and a half hour exam in May/June: 80%|
Many of the topics in this course have their starting point in results which hold for finite-dimensional spaces and in the geometric properties of Euclidean space. We will see that the natural generalisation of a finite-dimensional inner product space, which was studied in MC241, is a Hilbert space. We study the ideas of minimal distance and best approximation; this enables us to obtain information about some complicated function by considering the best approximation to it by simpler functions (usually polynomials) whose structure and properties are well-known. A second application of the theory of Hilbert spaces is in the study of orthogonal complements; this extends the idea of the Cartesian coordinate system, whereby we use two perpendicular axes in order to describe every point in the plane.
To know the definitions of and understand the key concepts introduced in this module.
To be able to investigate whether or not a linear operator between normed vector spaces is bounded, and determine its norm.
To be able to determine whether or not a normed vector space is an inner product space and whether or not it is complete.
To be able to prove the main results of this module and use them to solve a variety of problems, in particular in relation to continuity and boundedness of linear operators between normed vector spaces, and to best approximations and orthogonality in a Hilbert space.
The ability to present arguments and solutions in a coherent and logical form.
The ability to use the techniques taught within the course to solve problems.
Linear operator, continuous linear operator, bounded linear operator, equivalence of continuity and boundedness for linear operators, operator norm, unbounded linear operator, B(X, Y) is a normed vector space with the operator norm, examples to investigate whether or not a linear operator is bounded and determine its norm, proof for a bounded linear operator and for all that ,norm of a composition of bounded linear operators, concept of equivalent norms, to know that all norms on a finite-dimensional vector space are equivalent, example of inequivalent norms.
Inner product space, every inner product space is a normed vector space, use of parallelogram law to show that the the converse does not hold, Cauchy sequence, convergent sequence, completeness of a normed vector space, Hilbert space, examples, every finite-dimensional normed vector space is complete, every finite-dimensional inner product space is a Hilbert space, Banach space, examples, fixed point, contraction mapping, contraction mapping theorem, construction of L2 as the completion of an inner product space.
Orthogonal elements and orthogonal set, orthonormal set, convex set, minimum distance theorem in a Hilbert space, orthogonality lemma, orthogonal complement, proof that if E is any subset of a Hilbert space H then is a complete subspace of H, Projection theorem, Bessel's inequality, Riesz representation theorem.
E. Kreysig, Introductory Functional Analysis with Applications, Wiley.
N. Young, An Introduction to Hilbert Space, CUP.