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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?

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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.

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. But I have not seen this implication derived explicitly.

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