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ℒᵖ space
Definition
| $ p\in [1,\infty) $ |
| $ \mathbb K = \mathbb C \lor \mathbb R $ |
| $ \langle X,\Sigma,\mu\rangle $ … measure space |
| $f\in\mathcal L^p(X,\mu)$ |
| $f:X\to \mathbb K $ |
| $\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ … finite |
Discussion
$\mathcal L^p(X,\mu)$ 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} $ |
Context
Subset of
Requirements
Parents
