# What is the $L^p_\text{loc}(\mathbb{R})$ topology?

I was reading some lecture notes that I found on internet and I saw something that I cannot completely understand. For $$p\geq 1$$, let's consider the $$L^p_\text{loc}(\mathbb{R})$$ space, which is given by $$L^p_\text{loc}:=\{f:\mathbb{R}\to\mathbb{R}: \, \hbox{ for all compact interval } \ I=[a,b], \ a I was wondering what is the "natural" $$L^p_\text{loc}$$-topology? What are open or closed sets in the "standard topology" given by $$L^p_\text{loc}$$. This question is trivial in the case of $$L^p$$ since we can use the $$L^p$$-norm to define the "natural" topology in $$L^p$$. However, in this case the $$L^p$$-norm makes no sense, since functions in $$L^p_\text{loc}$$ don't belong to $$L^p$$. Am I missing something? Maybe the natural topology is related to some weak or weak-* topology? If that is the case, is it true that $$(L^p_\text{loc})^*=L^q_\text{loc}$$ with $$\tfrac{1}{p}+\tfrac{1}{q}=1$$ as in the standard case $$L^p$$ and $$L^q$$ with $$p\neq \infty$$.

• It is the local p convergence, also called the p convergence on all compacts Apr 22, 2021 at 8:04
• basically, f_n -> g iff it converges to g in p-norm on all compacts Apr 22, 2021 at 8:06
• This is a typical example of a Fréchet space. Apr 22, 2021 at 18:29

A natural metric on this space is defined by $$d(f,g)=\sum_n \frac 1 {2^{n}} \min\{(\int_{-n}^{n} |f(x)-g(x)|^{p} dx)^{1/p} ,1\}$$.

In this topology $$f_k \to f$$ iff $$\int_{-n}^{n} |f_k(x)-f(x)|^{p}dx\to 0$$ for each $$n$$ iff $$\int_{a}^{b} |f_k(x)-f(x)|^{p}dx\to 0$$ whenever $$a iff $$\int_K |f_k(x)-f(x)|^{p}dx\to 0$$ for any compact set $$K$$.

The natural convergence on this space is the local p-convergence, defined by

$$f_n \to g \text{ iff } \forall C compact, {f_n}_{|C} \to_p g_{|C}$$

As stated by the other answer, this is a metric topology on $$\mathbb{R}$$ (because there is a countable cover of $$\mathbb{R}$$ by compact sets)

The reason why it is called "local" is because, by Borel-Lebesgue theorem, it is equivalent to the topology

$$f_n \to g \text{ iff } \forall x \in \mathbb{R} \exists V \text{ neighborhood of x}, {f_n}_{|V} \to_p g_{|V}$$