# The space of $k$ differential forms as a Fréchet space

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.

• Wouldn't a metric on TM give one on the associated bundles, for instance the dual and tensor powers of it? Then there's no need to look at coordinate charts. Changes of coordinates, if the set U is precompact, shouldn't change the topology, if you want to use this technique. – David Roberts Jul 10 at 21:34
• Is it because $U_I$ and $\bar U_I$ under two coordinates differ by a factor of smooth function on $U$, then on a precompact set, the two seminorms are bounded by each other thus equivalent? @DavidRoberts – Keith Jul 10 at 21:58
• @keith: this is true if $M$ is compact – Thomas Rot Jul 10 at 22:11
• If I remember correctly, these things are discussed here: staff.science.uu.nl/~crain101/AS-2013/main.pdf – Thomas Rot Jul 10 at 23:26
• @Keith that's what I'm thinking. You can get an upper bound for the Jacobian of the change of coordinates over the overlap, and so get an equivalence of seminorms. – David Roberts Jul 11 at 1:09