A special subspace of weighted spaces of holomorphic functions on the upper half plane

In this paper, we intend to define and study concepts of weight and weighted spaces of holomorphic (analytic) functions on the upper half plane. We study two special classes of these spaces of holomorphic functions on the upper half plane. Firstly, we prove these spaces of holomorphic functions on the upper half plane endowed with weighted norm supremum are Banach spaces. Then, we investigate these spaces of holomorphic functions on the upper half plane from a new aspect which has not been considered up to now. Indeed we prove that without any necessary condition on a weight such as restricting the rate of growth from below or above (constructing the upper bound or lower bound) or limit condition (except the continuity on the upper half plane) any weighted spaces of holomorphic functions on the upper half plane has a special subspace which can be written as countable intersection of closed sets.