SCHWEITZER 2002 Problem 9

Let $M$ be a connected, compact $C^{\infty}$ -differentiable manifold, and denote the vector space of smooth real functions on $M$ by $C^{\infty}(M)$ . Let the subspace $V\le C^{\infty}(M)$ be invariant under $C^{\infty}$ -diffeomorphisms of $M$ , that is, let $f\circ h\in V$ for every $f\in V$ and for every $C^{\infty}$ -diffeomorphism $h\colon M\rightarrow M$ . Prove that if $V$ is different from the subspaces $\{ 0\}$ and $C^{\infty}(M)$ then $V$ only contains the constant functions.