4
$\begingroup$

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.

  • 2
    If your intention is $\|f\|_n = \left(\int_{-n}^{n} |f|^{\color{red}{2}}\right)^{\color{red}{1/2}}$ then you're correct. The way you write it you'd get the topology of $L_{\operatorname{loc}}^{\color{red}{1}}(\mathbb{R}) \supset L_{\operatorname{loc}}^2(\mathbb{R})$. For general $\Omega$ just exhaust it by compact sets such that $K_{n+1}$ contains $K_{n}$ in its interior and check that the topology does not depend on the exhaustion.2011-09-18
  • 0
    @t.b. Thank you very much for the hints.2011-09-19
  • 0
    @t.b.:something not on mathemmatics:how did you edit to get the equation concerning $||f||$ in a single line?And how did you get my question into several paragraphs?(this looks indeed nice).2011-09-19
  • 0
    To get a formula displayed like this $$f(x) = \int_{0}^x e^{-t}\,dt$$ simply enclose it in double dollar signs `$$f(x) = \int_{0}^{x} e^{-t}\,dt$$`. To get paragraphs you need to hit enter twice. You can see what I did in detail by clicking [here](http://math.stackexchange.com/posts/65496/revisions) (you get there by clicking on edited [xx time ago](http://math.stackexchange.com/posts/65496/revisions) above my user name next to yours).2011-09-19
  • 0
    @t.b.:thanks again!:)2011-09-20
  • 0
    A keyword to search for is «final topology». The general nonsense attached to that is extremely useful when dealing with this situation and —in my experience— intuitively natural.2012-01-10

1 Answers 1