Lp space

Summary

For $1 \le p < \infty$, the space $L^p(\mu)$ consists of the measurable functions $f$ on a measure space with $\int \abs{f}^p \, d\mu < \infty$. Two functions that agree almost everywhere count as the same element. The norm is $\norm{f}_p = (\int \abs{f}^p \, d\mu)^{1/p}$, and with this norm $L^p(\mu)$ is a Banach space. The case $p = 2$ is a Hilbert space. These spaces are the basic function spaces of analysis, probability and partial differential equations.