Integration on Polynomials is Linear Operator

Theorem

Let $P \left({\R}\right)$ be the vector space of all polynomial functions on the real number line $\R$.

Let $S$ be the mapping defined as:

$\displaystyle \forall p \in P \left({\R}\right): \forall x \in \R: S \left({p \left({x}\right)}\right) = \int_0^x p \left({t}\right) \mathrm d t$

Then $S$ is a linear operator on $P \left({\R}\right)$.

Proof

Proved in Linear Combination of Integrals.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$: Example $28.6$