Norms on direct sums and tensor products

We first consider the construction of a norm on a direct sum of normed linear spaces and call a norm absolute if it depends only on the norms of the component spaces. Several clharacterizations are given of absolute norms. Absolute norms are then used to construct norms on tensor products of normed linear spaces and on tensor products of operators on normed linear spaces. 1. Introduction. In this paper, we consider the construction of norms on composite linear spaces formed from direct sums and tensor products of normed linear spaces and we consider properties of norms of operators on these spaces. The notion of an absolute norm is introduced as a natural generalization of the relatively familiar idea of an absolute vector norm on the space Cn of ordered n-tuples of complex numbers. Such norms on Cn correspond to the "coordinatewise symmetric" gauge functions as described by Ostrowski (3), and it is shown that our absolute norms on composite spaces correspond in a one-to-one fashion with the absolute vector norms on Cn. We are particularly interested in operator norms for which, in an appropriate sense to be detailed later,