Perturbation - Knowlpedia

For a reference element $u_0$ in a normed vector space, a perturbation is an increment $h$ , giving the modified object $u=u_0+h$ . A family is a small perturbation in that norm if $\|h_\varepsilon\|\to0$ ; a quantitative version is $\|h_\varepsilon\|=O(\varepsilon^a)$ with $a>0$ .

The chosen topology matters

For functions, small values do not imply small derivatives. For example, $\varepsilon\sin(x/\varepsilon)$ tends uniformly to zero while its first derivative does not. Perturbation arguments must name the norm, derivative seminorms, or weighted estimates used. An increment can be small relative to a growing reference field without being absolutely small.