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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 $$ L^p_\text{loc}:=\{f:\mathbb{R}\to\mathbb{R}: \, \hbox{ for all compact interval } \ I=[a,b], \ a<b, \ \Vert f\Vert_{L^p(I)}<\infty\}. $$ 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 $p\neq \infty$.

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

2 Answers 2

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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 $a <b$ iff $\int_K |f_k(x)-f(x)|^{p}dx\to 0$ for any compact set $K$.

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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

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

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