= Derivative seminorms on a compact interval
{title2=$p_k(f)=\max_{0\leq j\leq k}\|f^{(j)}\|_\infty$}
= Derivative seminorm
{synonym}
Here $C^\infty([0,1])$ means the real- or complex-valued <functions> <continuous> on $[0,1]$, <smooth> on $(0,1)$, with every interior <derivative> having a <continuous> extension to the endpoints; endpoint values of <derivatives> are these extensions. On this <space of smooth functions on a compact interval>, define
$$
p_k(f)=\max_{0\leq j\leq k}\sup_{t\in[0,1]}|f^{(j)}(t)|,
\qquad k=0,1,2,\ldots,\qquad f^{(0)}=f.
$$
These finite-valued <seminorms> increase with $k$ and separate points. In fact every $p_k$ is a <norm>, since it includes $p_0(f)=\|f\|_\infty$. The terminology <seminorm> includes <norms>; it does not assert that every measurement has a nontrivial <kernel of a seminorm>. <Smooth convergence> means $p_k(f_n-f)\to0$ for every fixed $k$, equivalently <uniform convergence> of every <derivative> separately. It does not demand one tolerance or one convergence index that works for all orders simultaneously. The maximum convention is equivalent to the <derivative supremum norm> convention using a finite sum, since $p_k(f)\leq\sum_{j=0}^k\|f^{(j)}\|_\infty\leq(k+1)p_k(f)$.
For a worked example, $f(t)=t^3$ gives $p_0(f)=1$, $p_1(f)=3$, $p_2(f)=6$, and $p_k(f)=6$ for $k\geq3$. The <linear functional> $\ell(f)=f''(1/2)$ obeys $|\ell(f)|\leq p_2(f)$ and is <continuous>. Similarly, <differentiation> is a <continuous linear operator> $D:f\mapsto f'$ for <smooth convergence>, because $p_k(Df)\leq p_{k+1}(f)$.
The <space of smooth functions on a compact interval> is complete for this <seminorm topology>. A <Cauchy sequence> has <uniform limits> $g_j$ for its <derivatives> of each order. Passing to the limit in the <fundamental theorem of calculus> identity
$$
f_n^{(j)}(t)-f_n^{(j)}(0)=\int_0^t f_n^{(j+1)}(s)\,ds
$$
gives $g_j(t)-g_j(0)=\int_0^t g_{j+1}(s)\,ds$, so $g_j'=g_{j+1}$ and $g_0$ belongs to the same space. This is the <completeness of the smooth-function metric> argument; the <metric> there and the increasing-$p_k$ <metric> above induce the same smooth <topology> and the same <Cauchy sequences>.
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