Duality for Spaces of Holomorphic Functions into a Locally Convex Topological Vector Space

Abstract

Consider an open subset O of the complex plane and a function f : O ÝÑ C. We understand the concept of holomorphicity of a complex-valued function. Suppose a function f defined on O takes values in a locally convex topological vector space (the complex plane is one particular example of a locally convex topological vector space). We want to study the notion of holomorphicity for f in this general setting. In the paper “Sur certains espaces de fonctions holomorphes I” [1], Alexandre Grothendieck deals with the concept of holomorphicity of vector-valued functions and duality. We read, understand, reproduce and at times fill in necessary details of the content of section 2 and a portion of section 4 of the paper. We understand the extension of Lebesgue integral to vector-valued functions on a measure space, named after I.M. Gelfand and B.J. Pettis as Gelfand-Pettis integral or Pettis integral (page 77 in [2]). Using the definition of Pettis integral and some results from complex analysis we introduce three notions of holomorphicity for the function f: holomorphicity of f on the open set O, weak derivability of f at a point zo P O and strong differentiability of f at a point zo P O. In Chapter 2 (section 2 in [1]) of this thesis, we study the conditions under which the first two notions of holomorphicity, i.e., f is holomorphic on the open set O and f is weakly derivable at every point of O coincide. Further the same conditions will imply the strong differentiability of f at each point of O. Also, we study Cauchy’s integral formula and Taylor’s expansion for vector valued holomorphic functions in Chapter 2. In section 4 of the paper “Sur certains espaces de fonctions holomorphes I” [1], for a subset of the Riemann sphere, Grothendieck introduces the space Pp ,Eq as the space of all locally holomorphic functions on vanishing at 8 if 8 P . He further considers two locally convex topological vector spaces E and F in separate duality and proves that the spaces Pp 1,Eq and Pp 2, Fq, where 1 and 2 are complementary subsets of the Riemann sphere, are in separate duality under some general conditions. In Chapter 3 of this thesis, which constitute the main portion of the thesis, we study the space Pp ,Eq and further present the argument of Grothendieck for separating duality with all details