$C^1([0 ,1]^n)$ is a normed vector space such that the underlying space is the set of all $C^1$ functions from $[0,1]^n$ to $\mathbb{R}$ and the norm is defined as $\|f\|=\sup |f| +\sum_{k=1}^n \sup |\partial_k f|$.
Are polynomials dense in $C^1([0 ,1]^n)$?
What I’ve tried: By the Stone-Weierstrass theorem, for any $\epsilon>0$, there are polynomials $p_1, …,p_n$ such that $\sup |\partial_k f-p_k|<\epsilon \quad (k=1,…,n)$. Let $P_k=\int_0^{x_k}p_k +f(0)$. Then $|f-P_k|=|\int_0^{x_k}(\partial_k f -p_k)|\leq \int_0^1|\partial_k f -p_k|\leq \epsilon$.