  The main goal of this paper is to study the topological properties of tensors
in tree-based Tucker format. These formats include the Tucker format and the
Hierarchical Tucker format. A property of the so-called minimal subspaces is
used for obtaining a representation of tensors with either bounded or fixed
tree-based rank in the underlying algebraic tensor space. We provide a new
characterisation of minimal subspaces which extends the existing
characterisations. We also introduce a definition of topological tensor spaces
in tree-based format, with the introduction of a norm at each vertex of the
tree, and prove the existence of best approximations from sets of tensors with
bounded tree-based rank, under some assumptions on the norms weaker than in the
existing results.
