I'm reading D.S.Jones' book, The theory of generalized functions, and in particular I'm studying theorem 3.18 on page 84, in the proof of which I don't understand a statement he makes. Before expressing the specific doubt, I need to provide some context to explain myself better. Let's start with two definitions.
A good function $\gamma(x)$ is defined as a function acting on reals, infinitely differentiable and such that it and all its derivatives are a $O\left(|x|^{-N}\right)$, for all $N$.
A generalized function is by definition a family of good functions $\{\gamma_n(x)\}_{n\in\mathbb{N}}$ such that the limit exists $\lim_{n\to\infty}\int_{-\infty}^\infty\gamma_n(x)\gamma(x)\mathrm{d}x$ for all good $\gamma(x)$.
I also report a theorem, without proving it (theorem 3.16 of the book).
$\forall\left\{\gamma_n(x)\right\}_{n\in\mathbb{N}}$, there exists $k$, $r$, and a constant $K$ such that $\left| \lim_{n\to\infty}\int_{-\infty}^\infty \gamma_n(x) \gamma(x) \mathrm{d}x\right| \leq K \max_{x\in\mathbb{R}}(1+x^2)^{k/2}\gamma^{(r)}(x)$.
Having said that, let's get to the question. In theorem 3.18 he defines a linear functional $F$ "on the space of good functions of the form $(1+x^2)^{k/2}\gamma^{(r)}(x)$, where $k$ and $r$ are those of theorem 3.16" (textual quote) in the following way:
$$F\left( (1+x^2)^{k/2}\gamma^{(r)}(x)\right) = \lim_{n\to\infty}\int_{-\infty}^\infty \gamma_n(x) \gamma(x)\mathrm{d}x$$
and says that by theorem 3.16 this functional has norm $K$.
The thing I don't understand is: is the domain of $F$ really a vector space? I don't see this, because if I add two good functions of that type, I get:
$$(1+x^2)^{k_1/2}\gamma_1^{(r_1)}(x) + (1+x^2)^{k_1/2}\gamma_2^{(r_2)}(x) \overset{?}{=}(1+x^2)^{k_3/2}\gamma_3^{(r_3)}(x)$$
and who would be then $k_3$, $r_3$ and $\gamma_3$? I might even try to say that $k_3=0$, $r_3=0$ and $\gamma_3 =(1+x^2)^{k_1/2}\gamma_1^{(r_1)}(x) + (1+x^2)^{k_1/2}\gamma_2^{(r_2)}(x) $, but then I could no longer exploit theorem 3.16 to be able to say that the norm of $F$ is $K$.