Given a smooth manifold $M$, the cotangent bundle $T^*M$ is dual to the tangent bundle $TM$ "fiberwise", i.e. $\forall x\in M$, $T^*_x(M)=(T_x(M))^*$.
Now, if the manifold is a vector space, then the tangent space is a topological vector space, and the cotangent bundle is homeomorphic to the topological dual of the tangent bundle, i.e. $T^*M \simeq (TM)^*$. My question is then two-fold:
Is the last assertion true (modulo some additional hypothesis that I forgot)? If yes, is it true in general for Fréchet manifolds or only for finite dimensional manifolds?
Are there situations other than this trivial one where it could be true? I would be interested on sufficient (and possibly necessary) conditions on $M$ such that: $TM$ is a topological vector space, and $T^*M \simeq (TM)^*$ (where $\simeq$ stands for homeomorphism).