Let $A$ be the disc algebra, i.e., $A=\{f\in C(\bar{U}):f \text{ is holomorphic in }U\}$, where $U$ is the unit disc in the complex plane. The norm considered is the supremum norm. Are there any established results about what its dual is?
Similarly, is there anything known about the dual of $C'([0,1])$, the space of continuously differentiable functions? The norm considered here is $\|f\|=\|f\|_{\infty}+\|f'\|_{\infty}.$
I would be grateful for some references on this, if any. Thank you.
The space $C'([0,1])$ is the direct sum of the subspace $\mathbb{C}\cdot 1$ of constant functions and the subspace $K$ of functions $f$ such that $f(0)=0$. Furthermore, $K$ is isomorphic to $C([0,1])$ via $f\mapsto f'$. It follows by the Riesz representation theorem that the dual of $C'([0,1])$ can be described as a direct sum $\mathbb{C}\oplus M([0,1])$, where $M([0,1])$ is the space of measures on $[0,1]$. Explicitly, every functional on $C'([0,1])$ can be written uniquely in the form $f\mapsto cf(0)+\int f' d\mu$, for some $c\in\mathbb{C}$ and some complex Borel measure $\mu$ on $[0,1]$.