dc.contributor.advisor | Bhattacharyya, Tirthankar | |
dc.contributor.author | Sehgal, Kriti | |
dc.date.accessioned | 2021-03-02T06:35:48Z | |
dc.date.available | 2021-03-02T06:35:48Z | |
dc.date.submitted | 2018 | |
dc.identifier.uri | https://etd.iisc.ac.in/handle/2005/4913 | |
dc.description.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 | en_US |
dc.language.iso | en_US | en_US |
dc.relation.ispartofseries | ;G29760 | |
dc.rights | I grant Indian Institute of Science the right to archive and to make available my thesis or dissertation in whole or in part in all forms of media, now hereafter known. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part
of this thesis or dissertation | en_US |
dc.subject | holomorphicity | en_US |
dc.subject | locally convex topological vector space | en_US |
dc.subject | Lebesgue integral | en_US |
dc.subject.classification | Research Subject Categories::MATHEMATICS | en_US |
dc.title | Duality for Spaces of Holomorphic Functions into a Locally Convex Topological Vector Space | en_US |
dc.type | Thesis | en_US |
dc.degree.name | MS | en_US |
dc.degree.level | Masters | en_US |
dc.degree.grantor | Indian Institute of Science | en_US |
dc.degree.discipline | Faculty of Science | en_US |