Drastic measures

Set

 context $F$ … set definiendum $S$ in it postulate $S:F\to{\mathbb K}\setminus\{0\}$ postulate $S$ … ${\mathbb K}$-linear
todo: ${\mathbb K}$-linear

Discussion

“Normalization w.r.t. $S$”,

$N_Sf:=(Sf)^{-1}\cdot f$,

has $SN_Sf=e$ and $[N_S]=[S^{-1}]$.

As $S$ is linear,

$N_S(c\,f)=N_S(f)$

We'll also write

$\bar{f}:=(Sf)^{-1}\cdot f$

Example 1

For $F$ being a set of functions from ${\mathbb N}$ to some monoid for which a sum is defined that's always invertible, the general case (I think) is

$Sf:=\sum_{n=0}^\infty (L_nf)(n)$,

where $(L_n)$ is a suitable sequence of linear operations (e.g. differential operators).

For $L_n={\mathrm{id}}$ we get the standard sum (see below).

Explicitly, let $a$ be a sequence to $\mathbb C$ and the sum is non-zero, then

$\bar{a}:{\mathbb N}\to[0,1]$

$\bar{a}(n):=\frac{a(n)}{\sum_{k=0}^\infty a(k)}$

has

$\sum_{n=0}^\infty \bar{a}(n)=1$

Example 1

For ${\mathbb D}\subseteq{\mathbb R}^m$ we have integrals.