Double algebraic genericity of universal harmonic functions on trees in the general case

TL;DR

This paper demonstrates that the set of universal harmonic functions on trees contains two dense linear subspaces intersecting only at zero, extending previous results to the most general case with functions valued in topological vector spaces.

Contribution

It introduces the existence of two dense, intersecting-only-at-zero linear subspaces of universal harmonic functions on trees in the general topological vector space setting.

Findings

Universal functions form two dense linear subspaces intersecting only at zero.

Results extend previous work to functions valued in topological vector spaces.

The study applies to the most general case of harmonic functions on trees.

Abstract

It has been shown that the set of universal functions on trees contains a linear subspace except zero, dense in the space of harmonic functions. In this paper we show that the set of universal functions contains two linear subspaces except zero, dense in the space of harmonic functions that intersect only at zero. We work in the most general case that has been studied so far, letting our functions take values over a topological vector space.