Lp空间

在数学中，L^p空间是由p次可积函数组成的空间；对应的ℓ^p空间是由p次可和序列组成的空间。它们有时叫做勒贝格空间，以昂利·勒贝格命名（Dunford & Schwartz 1958，III.3），尽管依据Bourbaki (1987)它们是Riesz (1910)首先介入。在泛函分析和拓扑向量空间中，他们构成了巴拿赫空间一类重要的例子。

L^p空间在工程学领域的有限元分析中有应用。

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Relations between p-norms

The grid distance or rectilinear distance (sometimes called the "Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:

${\displaystyle \left\|x\right\|_{2}\leq \left\|x\right\|_{1}.}$

This fact generalizes to p-norms in that the p-norm ||x||_p of any given vector x does not grow with p:

||x||_p+a ≤ ||x||_p for any vector x and real numbers p ≥ 1 and a ≥ 0. (In fact this remains true for 0 < p < 1 and a ≥ 0.)

For the opposite direction, the following relation between the 1-norm and the 2-norm is known:

${\displaystyle \left\|x\right\|_{1}\leq {\sqrt {n}}\left\|x\right\|_{2}.}$

This inequality depends on the dimension n of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.

In general, for vectors in Cⁿ where 0 < r < p:

${\displaystyle \left\|x\right\|_{p}\leq \left\|x\right\|_{r}\leq n^{(1/r-1/p)}\left\|x\right\|_{p}.}$