Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$.



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> **Question:** Is the space $C^\nu(X,Y)$ a ($C^\infty$ or $C^k$ for some $k \in \mathbb{N}$) Banach manifold which is *Lindelöf* (i.e. every open cover has a countable subcover)?
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Any help, as well as references are very much appreciated.




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Some motivation:  

My goal is to apply a parametric transversality theorem, where my parameter space is $C^\nu(X,Y)$. In order to so it is required that the parameter space is Lindelöf and a ($C^k$-) Banach manifold. I don't have much experience with function spaces and their topology, especially when the target is infinite dimensional. So I apologize if it turns out that my question is "well known".



Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?
See the paper 


* MR0226681 (37 #2268) 
Elĭasson, Halldór I.
Geometry of manifolds of maps. 
J. Differential Geometry 1 1967 169–194.


for a description of the manifold structure on $C^\nu(X,Y)$. But you need somthing like a Riemannian exponential mapping on $Y$ which is a diffeomorphism from a neighborhood of the 0-section in $TY$ to a neighborhood of the diagonal in $Y\times Y$; you can forget some properties of the exponential mapping, then this is called a local addition (see [here](http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf)).
If $Y$ is nice enough (smoothly paracompact or at least $C^1$-paracompact allows to construct a linear connection on $TY$; or you may be given a linear connection on $TY$) then you have this.



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> Edit: 
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If $X$ is compact and $Y$ is a separable Banach space (added in second edit), then $C^\nu(X,Y)$ is separable; In fact it is 
$C^\nu(X)\hat{\hat\otimes} Y$ for the completed inductive tensor product.
(see [here](http://www.mat.univie.ac.at/~michor/banach_modules_and_functors.pdf), e.g)
Then it is obvious that the tensor product is separable. 



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> Second edit: About $C^k$-mapping manifolds.
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In "Kriegl, Michor: The convenient setting ..."
(see my homepage) 14.11.1 states that there does not exist any Frechet differentiable bump function. You can embed an interval in any manifold, thus $C^0(M)$ does not admit differentiable bump functions.
Furthermore, $C^0$ sits at the top of $C^k(M)$, so also $C^k(M)$ does not admit differentiable bump functions. On the other hand, $C^\infty(M)$ is nuclear Frechet, so has smooth partitions of unity.
This is the reason why I vastly prefer $C^\infty$, and Sobolev completions, if I have to solve equations. 


