 Continuously differentiable functions on a compact interval form a Banach algebra

With complex scalars and pointwise multiplication, the product rule gives a submultiplicative norm. For a Cauchy sequence, uniform limits $f$ of the functions and $g$ of their derivatives satisfy $f(t)-f(0)={\int }_0^{t}g(s)ds$, proving $f^{\prime }=g$ and completeness. Polynomials are dense in this norm: approximate $f^{\prime }$ uniformly by polynomials and integrate, fixing the initial value. This also identifies the character space of C1 on a compact interval.