TITLE:
Space Topologies and Their Dual Space Topologies for Conventional Functional Space Topologies
AUTHORS:
Mahjoub A. Elamin
KEYWORDS:
Linear Forms, Dual Spaces, Frechet Spaces, Partial Derivatives, Distributions, Topological Structure, Weak Topology, Strong Topology
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.3,
March
25,
2024
ABSTRACT: In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topological vector spaces and it is in this context that we have chosen to present this work. We are interested in the topology of its spaces and in the topologies of their dual spaces. The first part, we presented the fundamental topological properties of topological vector spaces. The second part, we studied Frechet spaces and particularly the space S(Rn) of functions of class C∞ on Rn which are as well as all their rapidly decreasing partial derivatives. We have also studied its dual S'(Rn) the space of tempered distributions. The last part aims to define a topological structure on an increasing union of Frechet spaces called inductive limit of Frechet spaces. We study in particular the space D(Ω) of functions of class C∞ with compact supports on Ω as well as its dual D' (Ω) the space distributions over the open set Ω.