Are trigonometric polynomials dense in $C_{2\pi}^m$? Let $m \in N$ be fixed and let $C_{2\pi}^m$ be a class of functions $f : R \rightarrow R$ of class $C^m$ and periodic with a period $2\pi$ with the following metric 
$$d(f,g)=\sum_{k=0}^m  \sup_{\{x \in R \}}|f^{(k)}(x)-g^{(k)}(x)|$$
for $f,g \in C_{2\pi}^{m}$. Is the set of trigonometric polynomials dense in this metric space?

Yes they are, because you can restrict to an interval $[0, 2 \pi]$ of function $f(0) = f(2 \pi)$, and use that they are dense in $C[0,2 \pi]$. Integration $k$-times and observing carefully what gets added will give you an affirmative answer. It is a continuous, surjective operator $C \mapsto C^k$.
