Some aspects of the theory of norms

AbstractLet V be a finite dimensional vector space. Motivated by theory or applications, one might want to consider different kinds of norms on V. In this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years. The paper consists of five sections. Section 1 concerns the conditions on two vectors x,y ∈ V satisfying ∥x∥≤∥y∥ for all ∥·∥ in a certain class of norms. Section 2 concerns the isometry groups of G-invariant norms, i.e., norms ∥·∥ that satisfy ∥g(x)∥ = ∥x∥ for all x ∈ V and for all g ∈ G, where G is a group of unitary (orthogonal) operators on V. Section 3 concerns G-invariant norms that satisfy some special properties. Section 4 concerns the best approximation(s) x0 ∈ T of y, where y ∉ T ⊆ V, with respect to different kinds of norms. Additional open problems, topics, and references are mentioned in Section 5