Comparability of Metrics and Norms in terms of Basis of Exponential Vector Space
Dhruba Prakash Biswas, Priti Sharma, Sandip Jana, Jens Schwaiger

TL;DR
This paper compares metrics within exponential vector spaces, explores their orderly dependence, and constructs numerous independent norms on linear spaces, revealing their topological non-equivalence.
Contribution
It introduces the concept of orderly dependence among metrics in exponential vector spaces and constructs many independent norms with incomparable topologies.
Findings
Collection of all metrics forms a topological exponential vector space.
Characterization of orderly independence via comparing functions.
Existence of many totally non-equivalent norms in infinite-dimensional spaces.
Abstract
In this paper, we shall compare two metrics in terms of orderly dependence, a notion developed in exponential vector space in the article 'Basis and Dimension of Exponential Vector Space' by Jayeeta Saha and Sandip Jana in Transactions of A. Razmadze Mathematical Institute Vol. 175 (2021), issue 1, 101-115. Exponential vector space, in short 'evs', is a partially ordered space associated with a commutative semigroup structure and a compatible scalar multiplication. In the present paper we shall show that the collection of all metrics on a non-empty set , together with the constant function zero , forms a topological exponential vector space. We shall discuss the orderly dependence of two metrics through our findings of a basis of in different scenario. We shall characterise orderly independence of two…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Theoretical and Applied Studies in Material Sciences and Geometry · Advanced Numerical Analysis Techniques · Mathematical Control Systems and Analysis
