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We first show that, for a fixed locally compact manifold $N,$ the space$L^2(S^1,N)$ has not the homotopy type odf the classical loop space$C^\infty(S^1,N),$ by two theorems:  - the inclusion $C^\infty(S^1,N) \subset L^2(S^1,N)$ is null homotopic if $N$ is connected,  - the space $L^2(S^1,N)$ is contractible if $N$ is compact and connected.  After this first remark, we show that the spaces $H^s(S^1,N)$ carry a naturalstructure of Fr\"olicher space, equipped with a Riemannian metric, whichmotivates the definition of Riemannian Fr\"olicher space.

Remarks on the geometry and the topology of the loop spaces $H^{s}(S^1, N),$ for $s\leq 1/2.$