Convex Optimization in Normed Spaces

Functional and convex analysis are closely intertwined. In this chapter we recall the basic concepts and results from functional analysis and calculus that will be needed throughout this book. A first section is devoted to general normed spaces. We begin by establishing some of their main properties, with an emphasis on the linear functions between spaces. This leads us to bounded linear functionals and the topological dual. Second, we review the Hahn-Banach Separation Theorem, a very powerful tool with important consequences. It also illustrates the fact that the boundaries between functional and convex analysis can be rather fuzzy at times. Next, we discuss some relevant results concerning the weak topology, especially in terms of closedness and compactness. Finally, we include a subsection on differential calculus, which also provides an introduction to standard smooth optimization techniques. The second section deals with Hilbert spaces, and their very rich geometric structure, including the ideas of projection and orthogonality. We also revisit some of the general concepts from the first section (duality, reflexivity, weak convergence) in the light of this geometry. For a comprehensive presentation, the reader is referred to [30] and [94].