I have started studying Sobolev spaces and I came across a space referred to as $H^1_{loc}$. I am not sure what the $loc$ subscript infers? What is it that makes this space different from $H^1$? Why would you every specify this space instead of $H^1$?
2026-04-13 21:16:01.1776114961
What is the difference between $H^1_{loc}$ and $H^1$?
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For J.L. Lions and others, $H^m_{loc}(\Omega)$ spaces are made of functions of $H^m(\Theta)$ where $\Theta$ represents any open subset such that $\bar{\Theta}\subset \Omega$. This allows to define a family of seminorms, the norms in $H^m(\Theta)$, and in the end these spaces are Frechet spaces.