| 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 |
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} $ |