Aims & Learning Objectives:
Aims - To introduce and study abstract spaces and general ideas in analysis, to apply them to examples, and to lay the foundations for the level 4 unit in functional analysis.
Objectives - 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.
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. Metric completion M of a metric space M. Real inner-product spaces. Hilbert spaces. Cauchy-Schwarz inequality, parallelogram identity. Examples: l2,L2[0,1]:=C[0,1]. Separability of L2. Orthogonality, Gram-Schmidt process. Bessel's inquality, Pythagoras' Theorem. Projections and subspaces. Orthogonal complements. Riesz Representation Theorem. Complete orthonormal sets in separable Hilbert spaces. Completeness of trigonometric polynomials in L2 [0,1]. Fourier Series.