# “Representability” II

As we have seen in the last problem, we can identify a compact smooth manifold $M$ with $\mathrm{hom}_{\Bbb{R}\mathrm{-alg}}(C^\infty(M),\Bbb{R})$. Give a similar identification between the tangent bundle $TM$ and  $\mathrm{hom}_{\Bbb{R}\mathrm{-alg}}(C^\infty(M),\Bbb{R}[x]/(x^2))$.

# “Representability”

Let $M$ be a compact smooth manifold. Show that any $\Bbb{R}$-algebra morphism $C^\infty(M)\to\Bbb{R}$ is of the form $\mathrm{ev}_p$ for some $p\in M$, where $\mathrm{ev}_p(f)=f(p)$.