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ℒᵖ_space [2013/09/05 22:24] nikolaj |
ℒᵖ_space [2014/03/21 11:11] |
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| - | ===== ℒᵖ space ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $ p\in [1,\infty) $ | | ||
| - | | @#88DDEE: $ \mathbb K = \mathbb C \lor \mathbb R $ | | ||
| - | | @#88DDEE: $ \langle X,\Sigma,\mu\rangle $ ... measure space | | ||
| - | | @#FFBB00: $f\in\mathcal L^p(X,\mu)$ | | ||
| - | |||
| - | | @#55EE55: $f:X\to \mathbb K $ | | ||
| - | | @#55EE55: $\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ ... finite | | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | $\mathcal L^p(X,\mu)$ is a seminormed $\mathbb K$-vector space with [[Pointwise 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} $ ^ | ||
| - | ==== Context ==== | ||
| - | === Subset of === | ||
| - | [[Seminorm]] | ||
| - | === Requirements === | ||
| - | [[Function integral]] | ||
| - | === Parents === | ||
| - | [[Pointwise function product]] | ||
