Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Author:
Kampoukou Angeliki
Supervisors info:
Βασίλης Νεστορίδης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Προβολικά όρια χώρων συναρτήσεων στις πολλές μιγαδικές μεταβλητές και ένα θεώρημα του Banach με εφαρμογή στους τόπους ολομορφίας.
Translated title:
Projective limits of function spaces of several complex variables and a theorem of Banach with application to domains of holomorphy.
Summary:
This dissertation deals with two issues. The first one is connected with the projective limit of spaces of holomorphic or harmonic functions defined on a sequence consisted of some open and bounded subsets of an open and connected subset Ω of ₵^d. Particularly, it is proven that the projective limit of complex function spaces, satisfying certain criteria, such as Bergman spaces, Hardy spaces, Nevanlinna class, BMOA spaces etc, is homeomorphic and linearly isomorphic to the space of holomorphic functions endowed with the topology of uniform convergence on compacta. Then the result is also proven for spaces of harmonic functions. The second issue concerns the non extendability of holomorphic functions defined on domains of ₵^d . It is proven that the set of non extendable holomorphic functions on some classes of functions F is a G_δ and dense subset of F. This proof is made with Baire's theorem and Montel's complex analysis theorem. This result cannot be proved in such a strong form, using a Banach's functional analysis theorem, with which it is proven that the set of all non expandable holomorphic functions is residual in F. Banach's theorem states that the continuous and linear image of a Fréchet space is either equal to the range or it is meager in the range, namely it is contained in a F_σ and thin subset of the range. However, the continuous and linear image of a Fréchet space can be of arbitrarily high Borel complexity.
Main subject category:
Science
Keywords:
projective limit, holomorphic functions, Fréchet spaces of holomorphic functions, extendability, domains of holomorphy, weak domains of holomorphy, analytic sets, Borel hierarchy
