Example

For $x$ real tending to $+\infty$ the periodic bounded functions on $\mathbf{R}$, having the same period $\tau$, satisfy conditions (CO₁), (CO₂) and (CO₃): if $\lim_{x \to +\infty} a(x) = 0$ then for every $\varepsilon > 0$ there is an $x_0$ such that $|a(x)| \leq \varepsilon$ for every $x \geq x_0$; one deduces that $|a(x)| \leq \varepsilon$ for $0 \leq x \leq \tau$ too, since there exists an integer $n$ such that $x + n\tau \geq x_0$, and such that $a(x) = a(x + n\tau)$; since $\varepsilon$ is arbitrary one has $a(x) = 0$ on $[0, \tau]$, hence everywhere.

With the notation of V, p. 222, we will say that $\sum_{\lambda \leq \alpha} a_\lambda g_\lambda$, where the $a_\lambda$ belong to $C$ and all but a finite number of them are zero, is an asymptotic expansion of $f$ with coefficients in $C$, to precision $g_\alpha$, if $f - \sum_{\lambda \leq \alpha} a_\lambda g_\lambda \ll g_\alpha$; for every index $\mu$ such that $a_\mu \neq 0$ then $a_\mu g_\mu$ is the principal part of $f - \sum_{\lambda < \mu} a_\lambda g_\lambda$, relative to $\mathscr{E}$ and to $C'$, which proves the uniqueness of the asymptotic expansion of $f$ (to precision $g_\alpha$) when it exists.

The methods given in n 3 (V, p. 223) for forming an asymptotic expansion for $f_1 + f_2$ or for $[f_1 . f_2]$, starting from given asymptotic expansions for $f_1$ and $f_2$, can again be applied to expansions with variable coefficients, so long as the $[a_\lambda . b_\mu]$ belong to the domain of coefficients $C$ corresponding to the normed space $V$ or admit an asymptotic expansion with coefficients in $C$.