Lp space

Lp空间

Analysis, Measure & PDE · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a measure space and 1<=p<infinity, Lp consists of measurable real or complex functions with finite integral of |f|^p, identified when equal almost everywhere, with norm (integral |f|^p)^(1/p). L-infinity instead uses the essential-supremum norm.

给定测度空间，1≤p<∞ 时，Lp 空间由 p 次绝对值可积函数的几乎处处相等类组成，范数为该积分的 p 次方根；p=∞ 时用本性上确界。

Example

For f(x)=xf(x)=x on [0,1][0,1], ∥f∥2=(∫01x2dx)1/2=1/3\|f\|_2=(\int_0^1x^2dx)^{1/2}=1/\sqrt3.

Usage and distinctions

The equivalence classes remove changes on null sets; literal pointwise functions would give only a seminorm.

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Definition sources

  1. John K. Hunter — Measure Theory§7.1, Definitions 7.1, 7.3–7.4 · Checked 2026-10-14

Translation sources

  1. 李东风（北京大学）— 金融时间序列分析备课笔记，附录EE.9.1 定义，L^p空间与本性有界函数 · Checked 2026-10-14
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