I am a beginner in differential geometry and would appreciate some pointers on how to answer the following question.
Let $M$ be a closed orientable Riemannian manifold with $\{e^1,...,e^n\}$ an oriented orthonormal basis of the cotangent bundle $T^* M$. My question is: how would one go about showing whether the operator $$c(e^k) : \Omega^*(M)\rightarrow\Omega^*(M),\,\,\omega\mapsto e^k\wedge\omega - i_{e^k}\omega$$ is bounded or not with respect to the usual $L^2$ inner product on $\Omega^*(M)$? Here $i_v$ denotes interior multiplication by $v$.
Thanks!
It's straightforward to see for a 1-form: take an orthonormal frame $e_1,\ldots,e_n$ and then $$|e_1\wedge\omega|^2=4\sum_{i<j}\Big((e_1\wedge\omega)(e_i,e_j)\Big)^2=4\sum_{j\geq 2}\Big((e_1\wedge\omega)(e_1,e_j)\Big)^2=\sum_{j\geq 2}\Big(\omega(e_j)\Big)^2$$ and $$|\iota_{e_1}\omega|^2=\Big(\omega(e_1)\Big)^2$$ so $|c(e_i)\omega|^2=|e_1\wedge\omega|^2+|\iota_{e_1}\omega|^2=|\omega|^2.$ The only extra difficulty for a $k$-form is in keeping track of all the normalization constants in defining norms and wedge products and such.