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} $

Parents

Subset of

Seminorm

Context

Function integral

Pointwise function product