Document Details

Document Type : Thesis 
Document Title :
Stone-Weierstrass type Theorem For non-archimedean Vector-valued Function spaces
نظريات من نوع ستون – ويرسترس لفراغات دالة متجه القيمة اللاأرخميدية
 
Subject : Mathematics 
Document Language : Arabic 
Abstract : In this thesis, we deal with Stone-Weierstrass type approximation theorems for continuous vector-valued functions in both the archimedean and non-archimedean settings. This theorem, first established by M.H. Stone in 1937 for the function spaces $C(X,\mathbb{R})$ and $C(X,\mathbb{C}),$ is a generalization of the classical Weierstrass approximation theorem of 1885 for the function space $C([0,1],\mathbb{R}).$ The first results in the non-archimedean area were proved by Dieudonne in 1944 and later by Kaplansky in 1949. We present the extensions of the Dieudonne-Kaplansky theorems to the function space $C(X,E)$ obtained by Prolla (1977, 1982) and Prolla-Verdoodt (1997) under the uniform, compact-open and strict topologies, where $X$ is a $0$-dimensional topological space and $E$ a topological vector space which is either non-archimedean or is over some non-archimedean valued field $\mathbb{F}.$ The approximation problem consists in finding the conditions on a $C(X)$-submodule $\mathcal{A}$ of $% C(X,E)$, so that $\mathcal{A}$ is dense in $C(X,E)$ in the above mentioned topologies. The key argument in the proofs is to use suitable lemmas on \textquotedblright partition of unity\textquotedblright . The last chapter contains some new results for the strict topology, where, in addition to the Stone-Weierstrass theorem, we give a characterization of maximal closed submodules and ideals in $C_{b}(X,E 
Supervisor : Dr. Liaqat Ali Khan 
Thesis Type : Master Thesis 
Publishing Year : 1427 AH
2006 AD 
Added Date : Wednesday, June 11, 2008 

Researchers

Researcher Name (Arabic)Researcher Name (English)Researcher TypeDr GradeEmail
لجين عبدالإله أبو الحمايلAbo Al-Hamayel, Loujain AbdalelahResearcherMaster 

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 23735.pdf pdfالمستخلص

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