SOME ASPECTS OF THE THEORY OF NORMS

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 is an element of V satisfying parallel to x parallel to less than or equal to parallel to y parallel to for all parallel to(.)parallel to in a certain class of norms. Section 2 concerns the isometry groups of G-invariant norms, i.e., norms parallel to(.)parallel to that satisfy parallel to g(x)parallel to = parallel to x parallel to for ah x is an element of V and for all g is an element of 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) is an element of T of y, where y is not an element of T subset of or equal to V, with respect to different kinds of norms. Additional open problems, topics, and references are mentioned in Section 5.