LINEAR ALGEBRA AND ITS APPLICATIONS Bound Norms

Recently Daniel and Palmer [2J have given an interesting application of a little-known lemma due to A. E. Taylor** [3J concerning the existence of an "orthogonal" basis for a finite dimensional (real or complex) normed vector space. Here we shall apply Taylor's lemma to obtain some simple results which have apparently not been noticed before. Thus we shall show that, given any bound norm (3, we can find a representation M so that, for each homomorphism T (linear transformation), the norm (3(T) is sandwiched between two easily computed sub additive functions of the matrix M(T) of T (Theorem 5.1), which do .not depend on M. Further we show that our results are the best possible: that is, there exists a bound norm for which our bounds for the norm are attained. Our point of view owes much to concepts independently introduced by Wielandt [4J and Bauer [1]. Wielandt has defined norms on the set of all (real or complex) matrices, while Bauer observed that we may consider categories of vector spaces and corresponding norms.*** The two approaches are similar; we are in the happy position of wishing to make use of both. Thus we shall begin with a category of vector spaces and represent the homomorphisms as matrices. Our categories are sets; it would, however, be possible to state our results in terms of the category of all (real or complex) vector spaces, which is a proper class. Since we shall