Aims

To introduce students to the basic ideas of Functional Analysis, an important area of research developed to study integral and differential equations. To introduce students to the notion of convergence and continuous transformations on vector spaces.

Module Summary

Linear Analysis, also known as Functional Analysis, grew out of efforts of many mathematicians in the late 19th and early 20th centuries to develop a rigorous framework for the study of integral and differential equations arising in the natural sciences. Building on various ideas familiar from earlier modules in Linear Algebra, Calculus and Complex Analysis, this course will cover several of the most important concepts in Functional Analysis, including norms and inner products on vector spaces, completeness of norms (which leads to the central concepts of Banach spaces and Hilbert spaces), and bounded linear operators between normed spaces. Wherever possible the abstract theory will be illustrated using concrete examples, and there will be a particular emphasis on the parallels between the new material and concepts familiar from earlier courses.

Outline Of Syllabus

Norms and inner products on vector spaces, the Cauchy-Schwarz inequality, orthogonality, Pythagoras' theorem, the parallelogram law. Convergence of sequences, Cauchy sequences, completeness, Banach spaces, Hilbert spaces, closed subspaces. Examples to include sequence spaces and spaces of continuous functions. Bounded linear operators, dual spaces, adjoints, invertibility and the spectrum.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

Note: the exam for MAS8702 is more challenging than the exam for MAS3702