Metric spaces, Banach spaces, Hilbert spaces

In this unit, we develop the theory of metric spaces, Banach spaces and Hilbert spaces. These are the foundations that support the models of modern physics, including general relativity, quantum mechanics, and optimisation; and are also essential for understanding stochastic phenomena, signal processing and data compression, Fourier analysis, differential equations, and numerical analysis. Topics covered include a basic introduction to metric spaces, topology in metric and Banach spaces, dual spaces, continuous linear mappings between Banach spaces, weak convergence and weak compactness in separable Banach spaces, Hilbert spaces and the Riesz representation theorem. Applications of these theories may include the contraction mapping theorem and its usage to prove the Cauchy-Lipschitz theorem (existence and uniqueness of solution to ordinary differential equations).