From Kreyszig's Introductory Functional Analysis with Applications:
Let $X = C[a,b]$ be the Banach space of all continuous real-valued functions on $J=[a,b]$ with norm defined by $\|x\| = \max_{t \in J} |x(t)|$.
A numerical process of integration is defined as: $(1)$ A set of $n+1$ real numbers $t_0^{(n)}, \dots, t_n^{(n)}$ such that $a \le t_0^{(n)}, \lt \dots \lt t_n^{(n)}$ and $(2)$ a choice of $n+1$ real numbers $\alpha_0^{(n)}, \dots, \alpha_n^{(n)}$ with $$f_n(x) = \sum_k^n \alpha_k^{(n)}x(t_k^{(n)})$$ defined as the numerical process of integration.
Then he defines convergence of a numerical process of integration as:
$f_n$ is convergent for an $x \in X$ if for that $x$, $$f_n(x) \rightarrow f(x)$$ where $f$ is defined by $\int_a^b x(t) dt$.
But what does $\lim_{n \rightarrow \infty} f_n$ mean? For each $n$ there's a choice of a particular $\alpha$ set and $t$ set. But what are the conditions for the $\alpha$ and $t$ set for this convergence to happen? Because to me this could be interpreted as "for ALL $\alpha$ and $t$ sets".
Does this instead mean as $n$ approaches infinity there EXISTS an $\alpha$ and $t$ set where this convergence happens?
Your definition of "numerical process of integration" is missing a crucial part: it should say that for each $n\in\mathbb{N}$ you have numbers $t^{(n)}_0, \dots t^{(n)}_n$ and $\alpha^{(n)}_0,\dots,\alpha^{(n)}_n$ with the stated properties. So, a single numerical process of integration consists of a specific choice of $\alpha$'s and $t$'s for each $n$ and so defines a specific function $f_n$ on $X$ for each $n$. We then say this specific numerical process of integration converges for $x$ if $f_n(x)\to f(x)$.