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ℒᵖ_space [2013/09/05 22:32] nikolaj |
ℒᵖ_space [2014/03/21 11:11] (current) |
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| ===== 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 ==== | ||
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| ^ $ \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]] | ||
