The algebraic properties of linear spaces and convex sets

Abstract

This short chapter gives the background from linear algebra and the basic facts about convex sets that are needed throughout the book. The first section describes the notation and the conventions that will be used for concepts from linear algebra. The substantive items concern dual spaces, dual bases, the second dual and adjoint linear maps. A large variety of facts about convex sets are needed in the later chapters. The definition of a convex set and some examples of such sets are already needed in Chapter 1, where norms are introduced. Therefore, the second section of this chapter introduces the idea of convexity via a series of examples and definitions. However, most of the important facts about convex sets ( e.g. the separation and support theorems) are more easily discussed using such topological notions as closed, bounded, interior and continuous linear mapping . Consequently, after dealing with topological ideas in Chapter 1, we return to a fuller study of convex sets in Chapter 2. Linear spaces Throughout the book X (and occasionally Y ) will be used to denote a finite dimensional vector space over the field ℝ of real numbers. The dimension of X will usually be d . Small Roman letters v,w,x,y,z will be used for vectors and small Greek letters α,β,γ for scalars. Subspaces of X will be denoted by capital Roman letters such as L, M ; other capital Roman letters ( e.g. B, C, K ) will be used for convex sets, and T will indicate a linear transformation between vector spaces.