在數學中，Lp空間是由p次可積函數組成的空間；對應的ℓp空間是由p次可和序列組成的空間。它們有時叫做勒貝格空間，以昂利·勒貝格命名（Dunford & Schwartz 1958，III.3），盡管依據Bourbaki (1987)它們是Riesz (1910)首先介入。在泛函分析和拓撲向量空間中，他們構成了巴拿赫空間一類重要的例子。
Lp空間在工程學領域的有限元分析中有應用。
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:
- ${\displaystyle \left\|x\right\|_{2}\leq \left\|x\right\|_{1}.}$
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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:
- ${\displaystyle \left\|x\right\|_{1}\leq {\sqrt {n}}\left\|x\right\|_{2}.}$
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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:
- ${\displaystyle \left\|x\right\|_{p}\leq \left\|x\right\|_{r}\leq n^{(1/r-1/p)}\left\|x\right\|_{p}.}$
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