Question

Suppose that T : C∞(R) −→ C∞(R) is a linear constant coefficient differential operator. In this problem, let’s prove that if g is a polynomial, we can always find a polynomial
f with T(f) = g.
Let d be any positive integer, and Pd the vector space of polynomials of degree ≤ d.
The vector space Pd is a subspace of C∞(R). If f is in Pd (and so also in C∞(R)) explain why T(f) is also in the subspace Pd