Let \(V\) be the vector space of all polynomial functions over the field of real numbers. Let \(a\) and \(b\) be fixed real numbers and let \(f\) be the linear functional on \(V\) defined by \[ f(p) = \int_a^b p(x)\,dx. \] If \(D\) is the differentiation operator on \(V\), what is \(D^t f\)?
Solution
For any polynomial \(p\), \[ (D^tf)(p)=f(Dp)=\int_a^b p'(x)\,dx. \] By the fundamental theorem of calculus, \[ (D^tf)(p)=p(b)-p(a). \] Thus \(D^tf\) is the linear functional \[ \boxed{p\longmapsto p(b)-p(a)}. \]