  In this work, we study non-rough norms in L(X,Y), the space of bounded linear
operators between Banach spaces X and Y. We prove that L(X,Y) has non-rough
norm if and only if X* and Y have non-rough norm. We show that the injective
tensor product of X and Y has non-rough norm if and only if both X and Y have
non-rough norm. We also give an example to show that non-rough norms are not
stable under projective tensor product. We also study a related concept namely
the small diameter properties in the context of L(X,Y)*. These results leads to
a discussion on stability of the small diameter properties for projective and
injective tensor product spaces.
