Given $l(f)$ a linear operator, if $l(F)$ integrates $1,x,...,x^n$ exactly, show that it integrates all polynomials of degree $\leq n$ exactly.
How would I even go about this? I was shown this proof as a takeaway in lecture and I have absolutely no idea on how to even start it?
It is not clear what you mean by 'integrates exactly', lets suppose it means that if $I(p) = \int_a^b p(x) dx$ for some $a,b$.
Then $l(f) = I(f)$ for the functions $x \mapsto x^k$, $k=0,..,n$.
Since $l$ and $I$ are linear it follows that $l(x \mapsto \sum_k a_k x^k) = \sum_k a_kl ( x \mapsto x^k ) = \sum_k a_k I ( x \mapsto x^k ) = I(x \mapsto \sum_k a_k x^k)$.