CHARACTERISATION OF EQUIVALENT NORMS ON A LINEAR SPACE USING EXPONENTIAL VECTOR SPACE
Print ISSN: 0972-7752 | Online ISSN: 2582-0850 | Total Downloads : 43
DOI: https://doi.org/10.56827/SEAJMMS.2023.1903.9
Author :
Dhruba Prakash Biswas (Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata-700019, INDIA)
Priti Sharma (Bangabasi College, 19, Rajkumar Chakraborty Sarani, Kolkata - 700009, INDIA)
Sandip Jana (Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata-700019, INDIA)
Abstract
In this paper we have found a necessary and sufficient condition for equivalence of two norms on a linear space using the theory of exponential vector space. Exponential vector space (`evs' in short) is an ordered algebraic structure which can be considered as an algebraic ordered extension of vector space. This structure is axiomatised on the basis of the intrinsic properties of the hyperspace $\mathscr{C}(\X)$ comprising all nonempty compact subsets of a Hausdorff topological vector space $\X$. Exponential vector space is a conglomeration of a semigroup structure, a scalar multiplication and a compatible partial order. We have shown that the collection of all norms defined on a linear space, together with the constant function zero, forms a topological exponential vector space. Then using the concept of comparing function (a concept defined on a topological exponential vector space) we have proved the aforesaid necessary and sufficient condition; also we have proved using comparing function that in an infinite dimensional linear space there are uncountably many non-equivalent norms.
Keywords and Phrases
Equivalence of norms, exponential vector space, topological exponential vector space, zero primitive evs, comparing function.
A.M.S. subject classification
46A99, 46B99, 06F99.
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