Searching for linear structures in the failure of the Stone-Weierstrass theorem

TL;DR

This paper explores the failure of the Stone-Weierstrass theorem by analyzing the large linear structures within the set of continuous functions that are not approximable by certain subalgebras, revealing nuanced lineability properties.

Contribution

It provides a detailed description of the lineability and codimension of non-approximable functions in the context of the Stone-Weierstrass theorem, including specific cases involving polynomials on the boundary of the disk.

Findings

Identifies the precise codimension of the closure of subalgebras in continuous functions.

Describes conditions under which large linear spaces exist outside the closure of these subalgebras.

Analyzes lineability properties of functions not approximable by polynomials on the boundary of the disk.

Abstract

We investigate the failure of the Stone-Weierstrass theorem focusing on the existence of large dimensional vector spaces within the set $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$, where $L$ is a compact Hausdorff space and $\mathcal{A}$ is a self-adjoint subalgebra of $\mathcal{C}(L, \mathbb{K})$ that vanishes nowhere on $L$ but does not necessarily separate the points of $L$. We address the problem of finding the precise codimension of $\overline{\mathcal{A}}$ in a broad setting, which allows us to describe the lineability of $\mathcal{C}(L, \mathbb{K}) \setminus \overline{\mathcal{A}}$ in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set $(\mathcal{C}(\partial{D}, \mathbb{C}) \setminus \overline{\text{Pol}(\partial{D})}) \cup \{0\}$, where $\text{Pol}(\partial{D})$ is the set of all complex polynomials in one variable restricted to the boundary of the unit disk. Recent lineability properties are also taken into account.