The first part of the course MAA302 is devoted to the theory of topological and metric spaces in an abstract setting, including the basic notions of continuity, completeness, compactness, and connectedness. We then shift our focus towards the space of continuous functions on a compact set, with the important theorems of Arzèla-Ascoli and Stone-Weierstrass, as well as towards Banach spaces, including the following fundamental results in functional analysis: the uniform boundedness principle, the open mapping theorem, and the closed graph theorem. The final part of the course is devoted to differential calculus on Banach spaces, studying the important results of the inverse function and implicit function theorems, and concluding with an abstract theory of optimization, with and without constraints.