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

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.

Solution 1:

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 useful except to demonstrate that the space is metrizable (with, perhaps, no canonical metric): $$ d(f,g)=\sum_{\ell\ge 1} 2^{-n} {\int_{|x|\le \ell} |f(x)-g(x)| \over 1+\int_{|x|\le \ell}|f(x)-g(x)|} $$

The assertion that it's not normable really is that it has no norm in_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 not normable (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 dual spaces...)

Solution 2:

As Paul Garrett rightly says, a metric on $L^1_{loc}$ is not very useful to understand its locally convex structure. The seminorms $\|f\|_n=\int\limits_{\{\|x\|\le n\}} |f(x)|dx$ are much more appropriate: A sequence or net $(f_\alpha)$ converges in $L^1_{loc}$ to $f$ if and only if $\|f_\alpha-f\|_n\to 0$ for every $n\in\mathbb N$. Assuming that there is a single norm $\|\cdot\|$ on $X=L^1_{loc}$ having the same convergent sequences would imply (just by abstract continuity arguments for vector spaces with seminorms) that $\|\cdot\|\le c \|\cdot\|_n$ for some $n\in\mathbb N$ and $c>0$ and that $\|\cdot\|_m \le c_m\|\cdot\|$ with constants $c_m$. This yields $\|\cdot\|_{n+1}\le \tilde c\|\cdot\|_n$ which is clearly wrong in the particular case by considering functions with support in $(n,n+1]$.