Aggrandization of Spaces of Holomorphic Functions Reduces to Aggrandization on the Boundary

We show that grand spaces of holomorphic functions may be equivalently defined in terms of aggrandization related only to the boundary. We base ourselves on recent studies of the so-called local aggrandization of Lebesgue spaces and extent this approach to the case of arbitrary Banach spaces of functions on metric spaces. We apply this approach to prove, in the case of Bergman and Bergman–Morrey spaces on the unit disk, that these grand spaces may be equivalently defined as grand spaces with weighted aggrandization on the boundary.