I'm currently managing to understand how to find a completion of Pre-Hilbert space $\mathcal H_0$ in Stein's real analysis. The textbook says the completion $\mathcal H$ has three properties, and the proof of the second one (ii) $(f,g)_0 = (f,g)$ whenever $f,g \in \mathcal H_0$ is written on the red line as follows.
This is where I'm stuck on. The following proof is what I've tried at most.
$|(f_n-f, g_n-g)| \le ||f_n-f||_H ||g_n-g||_H$, and since $||f_n-f||_H \to 0$ and $||g_n-g||_H \to 0$ as $n \to \infty$, $|(f_n-f, g_n-g)| \to 0$. Thus $lim_{n \to \infty}(f_n,g_n) = (f,g)$.
This is what I've interpreted the definition of $(f,g)$ written on the red line. If this is correct, what should I do to induce $(f,g)_0$? The above proof doesn't include no inner product of $\mathcal H_0$, $(,)_0$, so I have a difficulty associating my trial with $(,)_0$.
Any help would be appreciated. Thank you.
I will write this as an Answer so the question can get closed.
Yes. It is just a typo.