Lp space

Remarks

Example: on $\R$ with Lebesgue measure, $f(x) = e^{-x^2}$ lies in $L^p$ for every $p$, since $\int e^{-p x^2} \, dx = \sqrt{\pi / p} < \infty$.

Non-example: $g(x) = 1 / (1 + \abs{x})$ on $\R$ is in $L^2$ but not in $L^1$. Indeed $\int_\R (1 + \abs{x})^{-1} \, dx = \infty$, while $\int_\R (1 + \abs{x})^{-2} \, dx = 2$.

The operations and the norm do not depend on the representatives. The map $\norm{\cdot}_p$ is a norm by the Minkowski inequality, and $\norm{f}_p = 0$ forces $f = 0$ almost everywhere. With this norm $L^p(\mu)$ is a Banach space, by the Riesz–Fischer theorem. For $p = \infty$ one uses the essential supremum instead, which gives the space L-infinity. With the counting measure on $\N$, $L^p$ is the sequence space $\ell^p$.