# Differences

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 ℒᵖ_space [2013/09/05 22:24]nikolaj ℒᵖ_space [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/05 22:32 nikolaj 2013/09/05 22:24 nikolaj 2013/09/05 22:23 nikolaj 2013/09/05 22:12 nikolaj created Next revision Previous revision 2013/09/05 22:32 nikolaj 2013/09/05 22:24 nikolaj 2013/09/05 22:23 nikolaj 2013/09/05 22:12 nikolaj created Line 1: Line 1: ===== ℒᵖ space ===== ===== ℒᵖ space ===== - ==== Definition ​==== + ==== Set ==== - | @#88DDEE: $p\in [1,\infty)$ | + | @#55CCEE: context ​    | @#55CCEE: $p\in [1,\infty)$ | - | @#88DDEE: $\mathbb K = \mathbb C \lor \mathbb R$ | + | @#55CCEE: context ​    | @#55CCEE: $\mathbb K = \mathbb C \lor \mathbb R$ | - | @#88DDEE: $\langle X,​\Sigma,​\mu\rangle$ ... measure space | + | @#55CCEE: context ​    | @#55CCEE: $\langle X,​\Sigma,​\mu\rangle$ ... measure space | - | @#FFBB00: $f\in\mathcal L^p(X,\mu)$ | + | @#FFBB00: definiendum ​| @#FFBB00: $f\in\mathcal L^p(X,\mu)$ | - | @#55EE55: $f:X\to \mathbb K$ | + | @#55EE55: postulate ​  | @#55EE55: $f:X\to \mathbb K$ | - | @#55EE55: $\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ ... finite | + | @#55EE55: postulate ​  | @#55EE55: $\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ ... finite | ==== Discussion ==== ==== 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 $\mathcal L^p(X,\mu)$ is a seminormed $\mathbb K$-vector space with [[Pointwise function product|pointwise]] addition and scalar multiplication and ^ $\Vert \cdot \Vert_p:​\mathcal L^p(X,​\mu)\to \mathrm R_+$ ^ ^ $\Vert \cdot \Vert_p:​\mathcal L^p(X,​\mu)\to \mathrm R_+$ ^ ^ $\Vert f\Vert_p:​=\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ ^ ^ $\Vert f\Vert_p:​=\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ ^ - ==== Context ​==== + ==== Parents ​==== === Subset of === === Subset of === [[Seminorm]] [[Seminorm]] - === Requirements ​=== + === Context ​=== [[Function integral]] [[Function integral]] - === Parents ​=== + === Related ​=== [[Pointwise function product]] [[Pointwise function product]]