Classically in complex analysis an analytic sheaf is a sheaf of modules over the structure sheaf O of rings. Motivated by problems in infinite dimensional holomorphy, we investigate stronger notions of analyticity which are suitable for the analysis of infinitely generated sheaves. The sheaves that we consider are analytic in the sense of Lempert-Patyi and analytic Frechet in the sense of Leiterer. In this thesis we demonstrate uniqueness of analytic structures on a certain class of O -modules. We also show that higher direct image sheaves do not always carry a suitable analytic structures. Along the way, we provide a few counter-examples to Grauert's Direct Image Theorem for infinitely generated analytic sheaves
