On very non-linear subsets of continuous functions

TL;DR

This paper extends the study of linear subspaces of continuous functions with unique maximum points, showing these results hold on general topological spaces and exploring conditions for their existence based on dimension.

Contribution

It generalizes previous results by proving the existence or non-existence of such subspaces on broad topological spaces, not just subsets of R.

Findings

Results hold for functions on general topological spaces.

Existence of subspaces depends on the desired dimension.

Certain subspaces cannot exist beyond specific dimension limits.

Abstract

In this paper we continue the study initiated by Gurariy and Quarta in 2004 on the existence of linear spaces formed, up to the null vector, by continuous functions that attain the maximum only at one point. Inserting a topological flavor to the subject, we prove that results already known for functions defined on certain subsets of R are actually true for functions on quite general topological spaces. In the line of the original results of Gurariy and Quarta, we prove that, depending on the desired dimension, such subspaces may exist or not.