  Tensor-based methods are receiving a growing interest in scientific computing
for the numerical solution of problems defined in high dimensional tensor
product spaces. A family of methods called Proper Generalized Decompositions
methods have been recently introduced for the a priori construction of tensor
approximations of the solution of such problems. In this paper, we give a
mathematical analysis of a family of progressive and updated Proper Generalized
Decompositions for a particular class of problems associated with the
minimization of a convex functional over a reflexive tensor Banach space.
