I am looking for the definition of the $C^1$ norm of a differential 1-form. I came across this while reading Theorem 10.1.1 from McDuff-Salamon's Symplectic Topology book. I am really stuck at this, and my guess so far has produced this, for a differential 1-form $\sigma$, and a given Riemannian metric $g$ on the symplectic manifold $M$, $$||\sigma||_{C^1}:= \inf_{x\in M} \sup_{||v||=1}|\sigma(v)|$$ where I am taking the infimum of the pointwise operator norms.
It would be really helpful if someone could confirm if this is correct, and provide with the correct definition of not.
I would also be interested to know if there are general formulas to define the $C^k$ norm of a differential $l$-form. Thanks.