= Continuously differentiable functions on a compact interval form a Banach algebra
{title2=$C^1[0,1],\quad\|f\|=\|f\|_\infty+\|f'\|_\infty$}
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 $f(t)-f(0)=\int_0^tg(s)\,ds$, proving $f'=g$ and completeness. Polynomials are dense in this norm: approximate $f'$ uniformly by polynomials and integrate, fixing the initial value. This also identifies the <character space of C1 on a compact interval>.
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