I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:

Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^k(M,N)$ of $C^k$-maps from $M$ to $N$, $k\in\mathbb{N}$, $k<\infty$. Then there exsists the structure of a smooth banach manifold on $C^k(M,N)$ and the underlying topology is the compact-open $C^k$-topology (the weak topology).

So far, I was only able to find references that prove more general results (and thus the proofs get harder and use things I don't really need for now), e.g. if $N$ is a banach manifold itself. These stronger results I found in:

• Eliasson, Geometry of Manifolds of Maps

• Abraham, Lectures of Smale on Differential Topology

Other sources I found deal only with $C^\infty(M,N)$ or $C^0(M,N)$. In this context the books of Lang (e.g. Fundamentals of Differential Geometry) are mentioned (e.g. here), but I'm not able to find these parts in his books.

I'd really appreciate it if someone could name me a reference where the above statement is proven.

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:

Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^k(M,N)$ of $C^k$-maps from $M$ to $N$, $k\in\mathbb{N}$, $k<\infty$. Then there exsists the structure of a smooth banach manifold on $C^k(M,N)$ and the underlying topology is the compact-open $C^k$-topology (the weak topology).

So far, I was only able to find references that prove more general results (and thus the proofs get harder and use things I don't really need for now), e.g. if $N$ is a banach manifold itself. These stronger results I found in:

• Eliasson, Geometry of Manifolds of Maps

• Abraham, Lectures of Smale on Differential Topology

Other sources I found deal only with $C^\infty(M,N)$ or $C^0(M,N)$. In this context the books of Lang (e.g. Fundamentals of Differential Geometry) are mentioned (e.g. here), but I'm not able to find these parts in his books.

I'd really appreciate it if someone could name me a reference where the above statement is proven.

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:

Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^k(M,N)$ of $C^k$-maps from $M$ to $N$, $k\in\mathbb{N}$, $k<\infty$. Then there exsists the structure of a smooth banach manifold on $C^k(M,N)$ and the underlying topology is the compact-open $C^k$-topology (the weak topology).

So far, I was only able to find references that prove more general results (and thus the proves get harder and use things I don't really need for now), e.g. if $N$ is a banach manifold itself. These stronger results I found in:

• Eliasson - Geometry of Manifolds of Maps

• Abraham - Lectures of Smale on Differential Topology

Other sources I found deal only with $C^\infty(M,N)$ or $C^0(M,N)$. In this context the books of Lang (e.g. "Fundamentals of Differential Geometry") are mentioned (e.g. here), but I'm not able to find these parts in his books.

I'd really appreciate it if someone could name me a reference where the above statement is proven.

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:

Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^k(M,N)$ of $C^k$-maps from $M$ to $N$, $k\in\mathbb{N}$, $k<\infty$. Then there exsists the structure of a smooth banach manifold on $C^k(M,N)$ and the underlying topology is the compact-open $C^k$-topology (the weak topology).

So far, I was only able to find references that prove more general results (and thus the proofs get harder and use things I don't really need for now), e.g. if $N$ is a banach manifold itself. These stronger results I found in:

• Eliasson, Geometry of Manifolds of Maps

• Abraham, Lectures of Smale on Differential Topology

Other sources I found deal only with $C^\infty(M,N)$ or $C^0(M,N)$. In this context the books of Lang (e.g. Fundamentals of Differential Geometry) are mentioned (e.g. here), but I'm not able to find these parts in his books.

I'd really appreciate it if someone could name me a reference where the above statement is proven.