Comparability of Metrics and Norms in terms of Basis of Exponential Vector Space

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 $\mathcal{D(\mathbf X)}$ of all metrics on a non-empty set $\mathbf X$, together with the constant function zero $O$, forms a topological exponential vector space. We shall discuss the orderly dependence of two metrics through our findings of a basis of $\mathcal{D(\mathbf X)}\smallsetminus\{O\}$ in different scenario. We shall characterise orderly independence of two elements of a topological evs in terms of the comparing function, another mechanism developed in topological exponential vector space, which can measure the degree of comparability of two elements of an evs. Finally, we shall discuss existence of orderly independent norms on a linear space. For an infinite dimensional linear space we shall construct a large number of orderly independent norms depending on the dimension of the linear space. Orderly independent norms are precisely those which are totally non-equivalent, in the sense that they produce incomparable topologies.