Can someone please help me to solve this problem:
For $r\in \mathbb{N}$ ,we note :
$E_r$={$P\in \mathbb{R}[x_1,x_2,...,x_d] $ such that $deg P\le r$ } And for $P\in E_r$ we define the function $R_P:\mathbb{R}^d \rightarrow \mathbb{R}$
by: $R_P(x)=\sum_{1\le|\alpha|\le r} |\partial_x^{\alpha}P(x)|^{\frac{1}{|\alpha|}} $
By using the Taylor formula for a polynômial given by:
$P(x_0+t)=\sum_{|\alpha |\le deg P} \frac{\partial_x^{(\alpha)} P(x_0)}{\alpha !} t^{\alpha }$
I want to show that the family of polynomials $(P(x_0+\frac{x}{R_P(x_0)})-P(x_0))_{x_0\in \mathbb{R}^d,P\in E_r}$ is relatively compact in $\mathbb{R}[x_1,x_2,...,x_d]$ with the $L^{\infty}(R^d)$ norme . Thanks