Linearization representation

Theorem

$X,Y$ … Banach spaces with topology
$f\in C(X,Y)$
postulate $ f(x+y) = f(x) + \int_0^1\ Df(x+t\ y)\cdot y\ \mathrm d t $

Discussion

$ f(x) = f(0) + \int_0^1\ Df(t\ x)\cdot x\ \mathrm d t $

Reference

Parents

Context

Fréchet derivative, Function integral