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ℒᵖ_space [2013/09/05 22:12] nikolaj created |
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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 ==== | ||
+ | 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 of === | + | === Context === |
[[Function integral]] | [[Function integral]] | ||
- | === Parents === | + | === Related === |
[[Pointwise function product]] | [[Pointwise function product]] |