![]() | Department of Mathematics | |||
![]() |
Credits: 10 | Convenor: Prof. S. C. Koenig | Semester: 2 (weeks 15 to 26) |
Prerequisites: | essential: MA1152(=MC147) | |
Assessment: | Coursework:Exam: 20:80% | Examination: 0% |
Lectures: | 18 | Problem Classes: | 5 |
Tutorials: | none | Private Study: | 52 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Surgeries: | none | Total: | 75 |
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.
The application of mathematical principles and concepts to new situations.
Written presentation of mathematical arguments in a coherent and logical form.
Use of techniques from the module to solve problems.
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 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.
Linear operator, continuous linear operator, bounded linear operator,
equivalence of continuity and boundedness for linear operators, operator
norm, unbounded linear operator, 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.
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 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 is any subset of a Hilbert space
then
is a complete subspace of
, Projection theorem, Bessel's
inequality.
Bryan P. Rynne and Martin A. Youngson, Linear Functional Analysis, Springer Undergraduate Mathematics Series.
E. Kreysig, Introductory Functional Analysis with Applications, Wiley.
N. Young, An Introduction to Hilbert Space, CUP.
![]() ![]() ![]() ![]() ![]() |
Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.