 The linearity of the map "f" ↦ "f"("T") follows from the convergence of the integral and that linear operations on a Banach space are continuous.We recover the polynomial functional calculus when "f"("z") = ∑0 ≤ "i" ≤ "m"  "ai zi" is a polynomial.To prove this, it is sufficient to show, for "k" ≥ 0 and "f"("z") = "zk", it is true that "f"("T") = "Tk", i.e.