Given a smooth manifold $M$, can define define seminorms on $\Gamma(U,\bigwedge^kT^{\ast}M)$ for $U$ a coordinate open set by the following: $p^{s}_L(u = \sum_{I}u_I dx_I) = \sup_{x \in M}\max_{|I|=p, \alpha \leq s}|D^{\alpha}u_I(x)|$. But this seminorm is not independent of the choice of the coordinates, then how should I interpret it?

All the references that I found do not mention this matter, but I think it might be the case that different coordinates induce the same topology but I do not see how to show this.