Lp norm

Intuitively, $L^p$ spaces are a generalization of finite dimensional Euclidean spaces. In particular, the $L^p$ norm generalizes the ${\ell }_p$ norm to infinite dimensional spaces. $L^p$ spaces are sometimes called Lebesgue spaces.

Formally, for a measure space $(\Omega ,\mathcal{F},\mu )$, the $L^p$ norm for $1\le p<\infty$ is defined as

$${\left|f\right|}_p={\int }_{\Omega }|f{|}^p\text{d}\mu .$$

The set of all functions $f$ such that ${\left|f\right|}_p$ is finite is an $L^p$ space. For $p=\infty$, we instead define $$ \norm{f}_\infty = \ess\sup |f|.

Lp norms and spaces are generalized by [[Orlicz norm|Orlicz norms and spaces]]. Notes: - For $0\leq p<1$ one can use the same definition, but the resulting "norm" is a not a true norm—it does not obey the triangle inequality. It is sometimes called a "quasi-norm". - $L^2$ is the only [[Hilbert space]] among all $L^p$ spaces.