My first question is a general question on Riesz Representation that is related, but somewhat independent of the second question.
Consider the Banach space $C^2_0(\mathbb{R},\mathbb{R})$ with usual norm of sum of sup norms over all derivatives. My understanding of Riesz Representation is that all bounded linear transformation on this space are of the form $$ f \mapsto \sum_{k=0}^2 \int_\mathbb{R}f^{(k)}(x)h_k(x)\,\mathrm{d}x $$ where each $h_k \in L^1$ and may be identically zero. Is this reasoning correct? My motivation for asking is the following:
Let $h \in L^1(\mathbb{R})$. Then the following are all linear transformations:
\begin{align*} T_1f&:= \int_\mathbb{R}f(x)h(x)\,\mathrm{d}x \\ T_2f&:= \int_\mathbb{R}f'(x)h(x)\,\mathrm{d}x \\ T_3f&:=T_1f+T_2f. \end{align*}
Assuming I haven't made any mistakes, that could only be the case if my first question is true. It would also follow that $\ker(T_i)$ would all be Banach spaces and $\ker(T_1)\cap \ker(T_2) \subset \ker(T_3)$ would be as well. It seems to me that $\ker(T_1)\cap \ker(T_2)$ would be quite small. Are there any theorems that cover this idea for a general $h$?