English

Normed linear spaces which are isometric to order unit spaces

Functional Analysis 2024-05-14 v2

Abstract

In this paper, we consider the linear direct sum of a real normed linear space with an order unit space and with a base normed space to obtain respectively a new order unit space and a new base normed space. As a consequence, we find that an 1\ell_1-space may be shown to be an order unit space. (However, not in the natural order.) Dually, an \ell_{\infty}-space may be shown to be a base normed space. To understand this aberration, we characterize the real normed linear spaces which are isometrically isomorphic to order unit spaces. We prove that other than the classical case of \ell_{\infty}, 1\ell_1 is also isometrically isomorphic to an order unit space whereas 2\ell_2 is not isometrically isomorphic to any order unit space.

Keywords

Cite

@article{arxiv.2306.06549,
  title  = {Normed linear spaces which are isometric to order unit spaces},
  author = {Anil Kumar Karn},
  journal= {arXiv preprint arXiv:2306.06549},
  year   = {2024}
}

Comments

11 pages

R2 v1 2026-06-28T11:02:06.251Z