	Department of Mathematics & Computer Science

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MA2121 Linear Analysis

MA2121(=MC243) Linear Analysis

Credits: 10

Convenor: Dr. J. Watters

Semester: 1, weeks 7-12

Prerequisites:	essential: MA2102(=MC241), MA2151(=MC240)
Assessment:	Coursework: 20%	One and a half hour exam in May/June: 80%

Lectures:	18	Problem Classes:	5
Tutorials:	none	Private Study:	52
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	75

Explanation of Pre-requisites

The concepts of continuity and convergent sequences from MA2151 are used in this course. From MA2102, the basic definitions and results concerning linear maps between vector spaces and the theory of finite-dimensional inner product spaces are required.

Course Description

This module introduces some of the central ideas in modern analysis and concentrates on the theory of normed vector spaces. Earlier modules have considered functions individually, but it is often more useful to consider the vector space of all functions with a particular property, for example, the set of all continuous functions. Much of the module is motivated by such examples and by the infinite-dimensional vector spaces of sequences of complex numbers.

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 MA2102, 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.

Aims

This module aims to introduce some of the central ideas of modern analysis through the study of bounded and unbounded linear operators on normed vector spaces and inner product spaces. Students should understand that the concept of a Hilbert space is fundamental to this study and appreciate its role in solving problems of best approximation and orthogonality.

Objectives

To develop an understanding of the importance of Hilbert spaces within the theory of normed vector spaces.

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.

Transferable Skills

The ability to apply taught principles and concepts to new situations.

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.

Syllabus

Normed vector space, examples, continuous mapping between normed vector spaces, the continuity properties of a norm.

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 $T : X\rightarrow Y$ and for all $x \in X$ that $\Vert Tx\Vert \leq \Vert T\Vert \Vert x\Vert$ , 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 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 $E^{\perp}$ is a complete subspace of , Projection theorem, Bessel's inequality, Riesz representation theorem.

Reading list

Background:

E. Kreysig, Introductory Functional Analysis with Applications, Wiley.

N. Young, An Introduction to Hilbert Space, CUP.

Details of Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the Summer exam period. The 20% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.