A Unified Polynomial Approximation Scheme for Classical Function Spaces

We prove polynomial density theorems for a broad class of classical function spaces on domains $\Omega \subset \mathbb{R}^n$, in both local (compact-open) and global (Banach) topologies. In the local regime we show that restrictions of polynomials are dense in $C^k(\Omega)$, $C^\infty(\Omega)$, and $L^p_{loc}(\Omega), W^{k,p}_{loc}(\Omega)$ for arbitrary open $\Omega$. In the global regime on bounded domains we re-prove the classical density results for $C^0(\bar{\Omega})$ (Stone-Weierstrass) and $L^p(\Omega)$. For higher-differentiable function spaces ($k \geq 1$), polynomial density depends on regularity of the domain. We show density in $C^k_b(\bar{\Omega})$ for $C^k$-extension domains (e.g. smooth boundary) and in $W^{k,p}(\Omega)$ for Sobolev-extension domains (e.g. Lipschitz boundary). The proofs are organized around a single approximation pipeline: extension, compact truncation, analytic regularization, and Taylor approximation. This yields straightforward, standard and unified arguments that make explicit the domain assumptions underlying several folklore polynomial-density statements; we also record counterexamples showing that such hypotheses are necessary for global approximation in strong norms.