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

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

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Seminorm

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Function integral

Pointwise function product

ℒᵖ space

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