Let $V$ be the vector space of all polynomial functions on the real field and let $f$ be the linear functional defined by
$$f(p)=\int_{0}^{1}p(x)dx.$$
If $D$ is the differential operator over $V$, what does $D^t(f)$ mean?
As we know, $f:V\to\mathbb{R}$ and $D:V\to V$, so $D^t$ is well defined by means of composition with itself over $V$. However, given that $f\in V^*$, I don't know how to understand the symbol $D^{t}f$.