Lp spaces on groups

Summary

The spaces $L^p(G)$ of a locally compact group $G$ are the Lp spaces of $G$ with respect to a left Haar measure. The group acts on them by translations, which preserve the norm. For $1 \le p < \infty$ this action is continuous, a fact used constantly in harmonic analysis. The case $p = 1$ gives the group algebra and the case $p = 2$ gives the Hilbert space of the regular representation.