  The main goal of this paper is to extend the so-called Dirac-Frenkel
Variational Principle in the framework of tensor Banach spaces. To this end we
observe that a tensor product of normed spaces can be described as a union of
disjoint connected components. Then we show that each of these connected
components, composed by tensors in Tucker format with a fixed rank, is a Banach
manifold modelled in a particular Banach space, for which we provide local
charts. The description of the local charts of these manifolds is crucial for
an algorithmic treatment of high-dimensional partial differential equations and
minimization problems. In order to describe the relationship between these
manifolds and the natural ambient space we prove under natural conditions that
each connected component can be immersed in a particular ambient Banach space.
This fact allows us to finally extend the Dirac-Frenkel variational principle
in the framework of topological tensor spaces.
