Trees, linear orders and G\^ateaux smooth norms

TL;DR

This paper establishes a necessary condition for Gâteaux smooth norms on C(T) spaces using a new linearly ordered set Z, and explores the relationship between Gâteaux smooth lattice norms and strictly convex dual norms.

Contribution

Introduces a novel ordered set Z to characterize Gâteaux smooth norms on C(T) and links Gâteaux smooth lattice norms to strictly convex dual norms.

Findings

Necessary condition for Gâteaux smooth norms on C(T)

Equivalence between Gâteaux smooth lattice norms and strictly convex dual norms

Analogous criteria for Gâteaux and Fréchet smooth norms

Abstract

We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a G\^ateaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fr\'echet smooth norms. In addition, we prove that if C(T) admits a G\^ateaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.