|Owning Department/School:||Department of Mathematical Sciences|
|Level:||Masters UG & PG (FHEQ level 7)|
|Supplementary Assessment:||MA40043 Mandatory Extra Work (where allowed by programme regulations)|
|Requisites:||Before taking this unit you must take MA20216 and take MA20218 and while taking this unit you must take MA30041|
To introduce and study abstract spaces and general ideas in analysis, to apply them to examples, and to lay the foundations for the MA4 unit in functional analysis.
By the end of the unit, students should be able to state and prove the principal theorems relating to uniform continuity and uniform convergence for real functions on metric spaces, compactness in spaces of continuous functions, and elementary Hilbert space theory, and to apply these notions and the theorems to simple examples.
Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets)
Topics will be chosen from the following: Uniform continuity and uniform limits of continuous functions on [0,1]. Abstract Stone-Weierstrass Theorem. Uniform approximation of continuous functions. Polynomial and trigonometric polynomial approximation, separability of C[0,1]. Total Boundedness. Diagonalisation. Ascoli-Arzelà Theorem. Complete metric spaces. Baire Category Theorem. Nowhere differentiable function. Completion of a metric space. Inner-product spaces. Hilbert spaces. Cauchy-Schwarz inequality, parallelogram identity. Examples. Orthogonality, Gram-Schmidt process. Bessel's inequality, Pythagoras' Theorem. Projections and subspaces. Orthogonal complements. Complete orthonormal sets in separable Hilbert spaces.
MA40043 is Optional on the following programmes:Department of Computer Science