First question is about the definition. Let $U$ be an open subset of $R^n$. Let $f$ be $k$ times continuously differentiable function on $U$. $C^k$ norm of $f$ is defined as sum of uniform norm of $i^{\text {th}}$ derivative($i\leq k$) if it is finite. I used to know $C^k(U)$ as just $k$ times differentiable functions on $U$, but at some notes $C^k(U)$ seemed to be the space of those functions whose $C^k$ norm can be defined(i.e. finite). Which one is correct?
I will proceed using the latter definition of "$C^k$ functions".
Let $M$,$N$ be $C^k$ manifold. Is there a notion of $C^k(M,N)$, the space of $C^k$ maps between them? for the simple case $N=\mathbb R$, for this notion to make sense I should check that finiteness of $C^k$ norm is preserved under $C^k$ diffeomorphism. say $\phi$ is a coordinate changing function. If there is an additional condition that derivatives of $\phi$ are bounded, it follows that this is true. But for general manifold coordinate change doesn't need to be this type, doesn't it?
If there is a notion of $C^k(M,N)$(or $C^k(M,\mathbb R)$), is there a $C^k$-norm generalizing the norm of $C^k(U)$?($U$ being an open subset of Euclidean space) I suspect there aren't, at least natural ones.
I would be thankful for explanation or references.
$C^k$ usually means that all $k$-th order derivatives not only exist but are also continuous. (So that $C^0$ stands for continuous functions.) Thus, after restricting to compacts, you have a well-defined $C^k$-norm.
There is no canonical norm on $C^k(M,N)$, you need an extra structure like a Riemannian metric or embedding to some Euclidean space. The easiest thing to do is to embed (properly!) $N$ in some $R^n$ and then use the $C^k$-norm on the space $C^k(M, R^n)$ provided $M$ is compact. Of course, the norm will depend on the embedding, but what you care, in fact, is the topology on $C^k(M,N)$ which is independent of our choices. If $M$ is not compact, you do not get a norm, only norms on restrictions of smooth maps to fixed compact subsets. However, this allows you to define a topology on $C^k(M,N)$ in the same way one defines uniform convergence on compacts, just taking higher derivatives into account.