  We consider a framework for the construction of iterative schemes for
operator equations that combine low-rank approximation in tensor formats and
adaptive approximation in a basis. Under fairly general assumptions, we obtain
a rigorous convergence analysis, where all parameters required for the
execution of the methods depend only on the underlying infinite-dimensional
problem, but not on a concrete discretization. Under certain assumptions on the
rates for the involved low-rank approximations and basis expansions, we can
also give bounds on the computational complexity of the iteration as a function
of the prescribed target error. Our theoretical findings are illustrated and
supported by computational experiments. These demonstrate that problems in very
high dimensions can be treated with controlled solution accuracy.
