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 $$\mathcal {C}(L, \mathbb {K}) {\setminus } \overline{\mathcal {A}}$$ C ( L , K ) \ A ¯ , where L is a compact Hausdorff space and $$\mathcal {A}$$ A is a self-adjoint subalgebra of $$\mathcal {C}(L, \mathbb {K})$$ C ( L , 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 $$\overline{\mathcal {A}}$$ A ¯ in a broad setting, which allows us to describe the lineability of $$\mathcal {C}(L, \mathbb {K}) {\setminus } \overline{\mathcal {A}}$$ C ( L , K ) \ A ¯ in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set $$(\mathcal {C}(\partial {D}, \mathbb {C}) {\setminus } \overline{\text {Pol}(\partial {D})}) \cup \{0\}$$ ( C ( ∂ D , C ) \ Pol ( ∂ D ) ¯ ) ∪ { 0 } , where $$\text {Pol}(\partial {D})$$ Pol ( ∂ 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.