Dual of the Banach space of $k$-times continuously differentiable functions. Let $C^k([0,1])$ denote the Banach space of $k$-times continuously differentiable functions $f:[0,1]\to \mathbb R$ with norm $$\|f\|_{C^k}:=\max_{i=0,\dots,k}\sup_{x\in [0,1]}|f^{i}(x)|.$$
I'm trying to understand its (continuous) dual space -- is there a nice characterisation of it and perhaps a reference where I can read about such things?

I'm also interested in the dual space of $C^r$ for $r>0$ non-integer ($\lfloor r\rfloor$-times continuously differentiable functions with $\lfloor r\rfloor^{\text{th}}$ derivative being $(r-\lfloor r\rfloor)$-Holder continuous if anyone could shed some light on this. Many thanks!
 A: This problem seems similar to the characterization of a dual of a Sobolev space with positives indices (cf Adams Sobolves Spaces, p. 49). 
Let $L$ be a linear functional on $C^1(\Omega):=\{ u\in L^p_{\rm loc}(\Omega)\text{ s.t. } D^\alpha u \in C^0 \text{ for } 0\leq|\alpha|\leq 1\} $. $\Omega$ a compact Hausdorff space.
We
set $N:=\sum_{0\leq|\alpha|\leq 1} 1$ and construct a projection $P$ from $C^1(\Omega)$ to a subspace $W$ of $C^0(\Omega)^N$,
defining 
$$
P(u):=(D^\alpha u)_{0\leq|\alpha|\leq 1}
$$
 which is an isometric isomorphism (II). $C^1(\Omega)$ is a separable Banach space endowed with the  $W^{1,\infty}(\Omega)$ norm. Thus $W$ is a closed subspace of $C^0(\Omega)^N$. Since $P$ is an II,  a linear functional $L^*$ on $W$ is defined as : $\forall z \in W, \; \exists u \in C^1(\Omega)$ s.t. $L^*(z)=L(P(u))$. By the Hahn-Banach Theorem there exists a norm preserving extension $\tilde{L}$ to $C^0(\Omega)^N$ of $L^*$. By the Riesz representation Theorem for measures there exists a vector measure $\mu \in M^1(\Omega)^N$ s.t. 
$$
{\tilde L}(u) = \sum_{0\leq|\alpha|\leq 1} \langle \mu_\alpha,u_\alpha\rangle_{M^1(\Omega),C^0(\Omega)},\quad \forall u \in C^0(\Omega)^N.
$$
Thus we obtain for $u\in C^1(\Omega)$, that 
$$
L(u)=L^*(Pu)=\tilde{L}(Pu)=\sum_{0\leq|\alpha|\leq 1} \langle\mu_\alpha,D^\alpha u\rangle_{M^1(\Omega),C^0(\Omega)}
$$
The extension from 1 to $k$ seems to me straightforward. So the two main ingredients is the projections $P$ and the representation theorems.
A: Hint for the case $k=1$: If $c \in \mathbb R$ and $\mu$ is a finite real Borel measure on $[0,1],$ then
$$\varphi(f) = cf(0) + \int_0^1 f'\, d\mu$$
is a bounded linear functional on $C^1([0,1]).$
