In fact I am reading the book of Ohsawa, Analysis of Several Complex Variables, and I came across this line on page 13,
... $L^{2}_\mathrm{loc}(\Omega)$ with respect to the topology induced by the $L^2$ convergence on compact sets.
Yet I am not clear how to induce a topology from. I have a guess. In the following I am considering the case $\Omega=\mathbb{R}$. Define $||f||_n=\int_{-n}^n |f|$ for $f\in L^2_\mathrm{loc}(\mathbb{R}), n=1,2,3,\ldots$. Put $$||f||=\sum\limits _{n=1}^\infty\frac{||f||_n}{1+||f||_n} \frac{1}{2^n}.$$ Then $||\cdot||$ is a metric for $L_\mathrm{loc}^2 (\mathbb{R}) $. Is it this topology? Or something else? Would someone be kind enough to give me some hints on this? Thank you very much.