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