===== ℒᵖ space =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ p\in [1,\infty) $ |
| @#55CCEE: context     | @#55CCEE: $ \mathbb K = \mathbb C \lor \mathbb R $ |
| @#55CCEE: context     | @#55CCEE: $ \langle X,\Sigma,\mu\rangle $ ... measure space |

| @#FFBB00: definiendum | @#FFBB00: $f\in\mathcal L^p(X,\mu)$ |

| @#55EE55: postulate   | @#55EE55: $f:X\to \mathbb K $ |
| @#55EE55: postulate   | @#55EE55: $\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ ... 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 function product|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]]
=== Related ===
[[Pointwise function product]]