While reading articles, Sometimes i see collection of all smooth loops as hilbert manifold(pre). Sometime i see this space as banach manifold. Sometime i see this sapce as nuclear frechet space.
Can we give all structure to loop space.
By loop space, i mean collection of all smooth map $\gamma: S^1\to M$ where $M$ is some Riemannian manifold. So precise question is Does there exists (pre)hilbert manifold strucuture on $C^\infty(S^1, M)$.