To introduce and study abstract spaces and general ideas in analysis, to apply them to examples, and to lay the foundations for the MA4 analysis units.
Learning Outcomes.
By the end of the unit, students should be able to state and prove the principal theorems relating to completeness, compactness, and dense sets in metric and normed spaces, and to apply these notions and theorems to simple examples.
Content.
Metric spaces and normed spaces, convergence and continuity. Examples. Completion of a metric space. Dense and nowhere dense sets, separable spaces, Baire category theorem. Equivalence of compactness and sequential compactness in metric spaces, relatively compact sets, Arzelà–Ascoli theorem. Uniform approximation of continuous functions, polynomial and trigonometric polynomial approximation. Further topics, which might include: the abstract Stone–Weierstrass theorem, existence of nowhere differentiable functions, connectedness and path connectedness.
