be viewed as Hilbert spaces by only considering maps of suitable Sobolev class. For example we can consider the space of all maps from the unit circle into a manifold This can be topologized via the compact open topology as a subspace of the space of all continuous mappings from the circle to that is, the free loop space of The Sobolev kind mapping space described above is homotopy equivalent to the free loop space. This makes it suited to the study of algebraic topology of the free loop space, especially in the field of string topology. We can do an analogous Sobolev construction for the loop space, making it a codimension Hilbert submanifold of where is the dimension of See also References . Contains a general introduction to Hilbert manifolds and many details about the free loop space. . Another introduction with more differential topology. N. Kuiper, The homotopy type of the unitary group of Hilbert spaces", Topology 3, 19-30 J. Eells, K. D. Elworthy, "On the differential topology of Hilbert manifolds", Global analysis. Proceedings of Symposia in Pure Mathematics, Volume XV 1970, 41-44. J. Eells, K. D. Elworthy, "Open embeddings of certain Banach manifolds", Annals of Mathematics 91 (1970), 465-485 D. Chataur, "A Bordism Approach to String Topology", preprint https://arxiv.org/abs/math.at/0306080 External links Hilbert manifold at the Manifold Atlas Differential geometry General topology Generalized manifolds Manifolds Nonlinear functional analysis Riemannian geometry Riemannian manifolds Structures on manifolds