  We introduce a framework for spline spaces of hierarchical type, based on a
parent-children relation, which is very convenient for the analysis as well as
the implementation of adaptive isogeometric methods. Such framework makes it
simple to create hierarchical \emph{basis} with \emph{control on the
overlapping}. Linear independence is always desired for the well posedness of
the linear systems, and to avoid redundancy. The control on the overlapping of
basis functions from different levels is necessary to close theoretical
arguments in the proofs of optimality of adaptive methods. In order to
guarantee linear independence, and to control the overlapping of the basis
functions, some basis functions additional to those initially marked must be
refined. However, with our framework and refinement procedures, the complexity
of the resulting bases is under control, i.e., the resulting bases have
cardinality bounded by the number of initially marked functions.
