Hahn-Banach type extension results for linear operators on asymmetric normed spaces

TL;DR

This paper extends the Hahn-Banach theorem to linear operators on asymmetric normed spaces, providing a direct proof of their injectivity characterized by the binary intersection property, adapting classical ideas to asymmetric contexts.

Contribution

It offers a direct proof linking injectivity of asymmetric normed spaces to the binary intersection property, adapting Nachbin's classical approach to the asymmetric setting.

Findings

Established equivalence between injectivity and BIP in asymmetric normed spaces.

Provided a direct proof of the injectivity characterization.

Extended classical Hahn-Banach extension results to asymmetric spaces.

Abstract

We present some results related to Hahn-Banach extension theorem for linear operators on asymmetric normed spaces. L. Nachbin, Trans. Amer. Math. Soc. 68 (1950), proved that a Banach space has the extension property for linear operators (a property also called injectivity) if and only if it has the Binary Intersection Property (BIP), meaning that every family of mutually intersecting closed balls has nonempty intersection. Its analog for quasi-metric spaces, called mixed BIP, was considered by Kemajou et al. Topology Appl. 159 (2012). The equivalence of mixed BIP to the injectivity of an asymmetric normed space was proved by Conradie et al., Topology Appl. 231 (2017), derived from some properties of the injective hull of a quasi-metric space. The aim of the present paper is to give a direct proof of this result by adapting Nachbin's ideas to the asymmetric case. Keywords: quasi-metric space, asymmetric normed space, injectivity, binary intersection property, hyperconvexity, Isbell-completeness, Isbell-convexity