Some aspects of the theory of norms

Abstract

Let 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) x 0 ∈ 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.