Hilbertian Analysis, Duality and Convexity

This chapter introduces several essential concepts in functional analysis and establishes the link between duality and geometry. We begin with a simple and powerful theory: the study of Hilbert spaces. These are the spaces for which we have an analogue of Pythagoras’ theorem, so we can use the methods of Euclidean geometry in infinite-dimensional spaces. We will study the properties of orthogonality and convexity and see how they intervene in the study of the topological dual.