I have the following question which I'm not sure on:
Denote $C^1\Big([0,1] \Big)$ the set of differentiablly continuous functions $f:[0,1]\rightarrow \mathbb{C}$. Let $\varphi:C^1\Big([0,1] \Big) \rightarrow \mathbb{C}$ be a linear functional such that:
$\vert \varphi(f)\vert \leq C_1\Vert f \Vert_\infty+ C_2\Vert f' \Vert_\infty$ for all $f\in C^1\Big([0,1] \Big)$, where $C_1,C_2$ are positive constants.
I want to show that there exists a complex Borel measure $\mu$ on $[0,1]$ and a constant $K\in \mathbb{C}$ such that:
$\varphi(f)=\int f' d\mu + K\cdot f(0)$ for all $f\in C^1\Big([0,1] \Big)$.
My attempt :
I know that linear functionals on $(C_c([0,1]),\Vert \cdot \Vert_\infty)$ are given by $\int g d\nu$ where $\nu$ is a Radon complex measure so I'm trying to use this fact. I know that $f=\int_0^xf'dt+f(0)$ for all $f\in C^1\Big([0,1] \Big)$, hence:
$\varphi(f)= \varphi \Big( \int_0^xf'dt \Big)+f(0)\cdot \varphi(1)$
And I define $K:=\varphi(1)$. And I want to show that $\varphi \Big( \int_0^xf'dt \Big)$ is of the desired form $\int f'd\mu$, but am having trouble putting it into action.
I'm not sure this is indeed a correct approach, and would appreciate hints or remarks explaining why this line of thought is wrong.
As you note the map $\phi: C^1\to C^0\oplus \Bbb C,\ f\mapsto (f', \, f(0)\, )$ is a vector-space isomorphism. Further $$\|f\|_\infty= \sup_x\left|\int_0^x f'(t)\,dt+f(0)\,\right|≤\|f'\|_\infty + |f(0)|,$$ from which it follows that the map is also topologically an isomorphism.
Now $\varphi\circ\phi^{-1}$ is a continuous functional on $C^0\oplus \Bbb C$, so use your knowledge of functionals on $C^0$ to see that it is of the form $(f, z)\mapsto I(f)+ K\,z$ where $I$ is a Radon measure on $[0,1]$ and $K$ a complex number.