$L^1_{\text{loc}}$, Frechet Space and norm-distance

functional-analysislp-spacesnormed-spaces

I was talking with a professor of mine, last week, and he started talking about the fact that "the space $$L^1_{\text{loc}}$$ is a Frechet Space, and hence you cannot put a norm into it. At most you can put a distance."

Can someone please clarify this passage to me a bit? Maybe with some example?
Or please give me some good reference where to read about those questions too.

Yes, $$L^1_{\mathrm loc}(\mathbb R^n)$$ is a Frechet space, with the countable collection of seminorms $$\int_{|x|\le B}|f(x)|$$. Really, the most important point (which requires a little proof) is that it is complete. The standard trick of rewriting a countable collection of seminorms into a single metric (not norm, in general!) is not so useful except to demonstrate that the space is metrizable (with, perhaps, no canonical metric): $$d(f,g)=\sum_{\ell\ge 1} 2^{-n} {\int_{|x|\le \ell} |f(x)-g(x)| \over 1+\int_{|x|\le \ell}|f(x)-g(x)|}$$