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.

## Best Answer

Filling out peoples' apt comments:

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 usefulexceptto demonstrate that the space is metrizable (with, perhaps, nocanonicalmetric): $$ d(f,g)=\sum_{\ell\ge 1} 2^{-n} {\int_{|x|\le \ell} |f(x)-g(x)| \over 1+\int_{|x|\le \ell}|f(x)-g(x)|} $$The assertion that it's not

normablereally is that it has no normin_which_it_is_complete. (Equivalently, there's no norm that induces the Frechet metric, which is complete.)It takes a little work, but is standard, to show that a topological vector space (with a given topology) is

notnormable (meaning giving the same topology). This issue is discussed in most not-completely-introductory functional analysis books, expressing the necessity of introducing "fancier" structures than just Hilbert and Banach spaces.(And to topologize the space of test functions to make them "complete", requires yet more effort, ... and then there are

dualspaces...)