Date of Award
Master of Science
Master of Mathematics
In reading any book concerning metric spaces one will find that proof of the n-dimensional Euclidean Spaces as a metric space is generally omitted} the author claiming that its proof is trivial. However, when one begins to do this proof he wonders if he really knows the meaning of the word trivial.
The problem of this paper is to show that the word trivial is validly used only in case certain striking and fundamental properties of the numbers are known and have been proven previously. In addition, we will impose several different metrics on the space, actually constructing different spaces, and investigate the possibility of getting the same solution to a problem on these different spaces. This will involve showing the relationship between these spaces both metrically and topologically.
Included in Chapter I are notations, definitions and basic theorems that will be used in the development of the proof. In Chapter II a study is made to show that an n-dimensional Euclidean space is a metric space under three given metrics denoted by d 1, d2» and d3. Chapter III involves the metrical and topological relationship of the given metrics and Chapter IV gives a brief summary of this study.
E. E. Thornton
Prairie View Agriculture And Mechanical College
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Date of Digitization
John B Coleman Library
City of Publication
Banks, D. R. (1969). On The Metrizability Of An N-Dimensional Euclidean Space. Retrieved from https://digitalcommons.pvamu.edu/pvamu-theses/1446