## ℒᵖ space

### Set

 context $p\in [1,\infty)$ context $\mathbb K = \mathbb C \lor \mathbb R$ context $\langle X,\Sigma,\mu\rangle$ … measure space
 definiendum $f\in\mathcal L^p(X,\mu)$
 postulate $f:X\to \mathbb K$ postulate $\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ … finite

### Discussion

Trivial remark: As explained in the notation section of the entry relation concatenation, the symbol $|f|^p$ denotes the function obtained by concatenation of the functions $f$ and $x\mapsto |x|^p$.

$\mathcal L^p(X,\mu)$ is a seminormed $\mathbb K$-vector space with pointwise addition and scalar multiplication and

$\Vert \cdot \Vert_p:\mathcal L^p(X,\mu)\to \mathrm R_+$
$\Vert f\Vert_p:=\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$