Let $M$ be a smooth, compact $n$-dimensional manifold and $C^k(M)$ the space of real valued $C^k$-maps on $M$.
I am looking for a definition of the norm $|\cdot|_k$ on $C^k(M)$ that induces the Whitney-topology.
My suggestion for a possible definition is:
Fix a Riemannian metric on $M$ and a finite open cover $\{U_j\}_{j=1}^N$ of $M$ such that $\forall 1\leq j\leq N$ there exists a family $\{v_1^j,\ldots,v_n^j\}$ of orthonormal vector fields on $U_j$. For each $j$ also fix such a family of vector fields. Given $f\in C^k(M)$ define $$ |f|_k:=\sup_{j}\sup_{x\in U_j}\sup_{\alpha:|\alpha|\leq k} |D^\alpha f(x)| $$ where $\alpha=(\alpha_1,\ldots,\alpha_n)$ is a multiindex, $$ D^\alpha f:=\underbrace{D_{v_1^j}\cdots D_{v_1^j}}_{\alpha_1}\cdots \underbrace{D_{v_n^j}\cdots D_{v_n^j}}_{\alpha_n}f $$ and $D_vf:=(df)(v)$. If $\alpha=(0,\ldots,0)$, then $D^\alpha f:=f$.
Is this a sensible definition of the norm $|\cdot|_k$?
Also, I'm not sure if different choices of Riemannian metrics, covers of $M$ and vector fields give equivalent norms.
References are greatly appreciated. Thanks for your help.
Pick any finite set $\def\X{\mathcal X} \X =\{X_i:1\leq i\leq m\}$ of vector fields on your compact manifold such that for all $p\in M$ the tangent space $T_pM$ is spanned by the set $\{X_i|_p:1\leq i\leq m\}$.
If $f\in C^k(M)$, define $$ \|f\|_\X = \max_i \|X_{i_1}\cdots X_{i_\ell}f\|_\infty $$ with the max taken over all sequences $i=(i_1,\dots,i_\ell)$ of all lengths $\ell\in\{0,\dots,k\}$ of indices taken from $\{1,\dots,m\}$. This is easily seen to be a norm.
Suppose $\def\Y{\mathcal Y} \Y =\{Y_j:1\leq j\leq t\}$ is another such set. There exists smooth functions $g_{i,j}$ such that $Y_j=\sum_{i}g_{i,j}X_i$ and using this one can compare $\|\cdot\|_\X$ and $\|\cdot\|_\Y$.