On the norms of quaternionic extensions of real and complex linear mappings

We study the connections between the norms of bounded operators on real and complex normed spaces, and with range in quaternionic spaces, and of their quaternionic extensions. We study some aspects of the relations between the quaternionization and the complexification of an inner product real space. The problems that arise when k-linear mappings are defined for the quaternionic case, are considered too.