Apologies in advance if this is well-known; a google search did not produce anything useful.


Let $(M,p)$ be a pointed real analytic manifold. Are the (free or pointed) loop spaces of continuous, smooth and analytic loops in $M$ all homotopy equivalent?



Homotopy type of continuous/smooth/analytic loop spaces?
Suppose $S$ and $M$ are smooth manifolds. For simplicity let us also suppose that $S$ is compact. Then the inclusion $C^\infty(S, M)\hookrightarrow C^0(S, M)$ is a weak equivalence. This is a "standard" fact that follows from the Whitney approximation theorem. A proof can be found the book of Hirsch. There is a discussion [here](https://math.stackexchange.com/questions/1794666/reference-request-inclusion-of-smooth-maps-into-continuous-maps-between-smooth).


If $S$ and $M$ are real analytic manifolds my guess is that an analogous result holds for the inclusion of the space real analytic maps, because [there exists an analogue of Whitney's approximation theorem for real analytic maps](https://mathoverflow.net/questions/203627/is-the-analytic-version-of-the-whitney-approximation-theorem-true). But I have not seen this implication derived explicitly.


