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In this paper we consider those linear transformations from one tensor product of vector spaces to another which carry nonzero decomposable tensors into nonzero decomposable tensors. We obtain a general decomposition theorem for such transformations. If we suppose further that the transformation maps the space into itself then we have a complete structure theorem in the following two cases: (1) the transformation is onto, and (2) the field is algebraically closed and the tensor space is adoi:10.2140/pjm.1967.23.613 fatcat:ujdanrayjren7owq4jkoxhmdy4