# Loop space: various manifold structure.

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 $C^\infty(S^1, M)$.

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The smooth loop space is dense in the various Hilbert manifolds so in that sense one could say that it is a pre-Hilbert manifold, but manifolds modelled on spaces that aren't complete are Really Bad (not all derivatives exist) so you should either work with a genuine Hilbert manifold (but which has some not-very-nice loops) or you should put the natural Frechet manifold structure on smooth loops, which maybe isn't so nice but at least all the loops are smooth. Often in the literature, smooth loops are used as a nice dense subspace of some class of loops but not as a manifold in its own right. – Loop Space Jun 21 '12 at 11:14