In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p -norm for finite-dimensional vector spaces.

We’ll complete our discussion of Lebesgue measure and integration today, finding the “complete space of integrable functions” that contains the space of continuous functions.

L^\infty 和 L^p 都是完备的赋范空间（ Banach 空间），但 L^p 可分， L^\infty 不可分。 l^p 空间. l^p 空间的元素： p 次方可和的数列全体 l^p=\left\{ x=\{\xi_k\}:\mathop{\sum}\limits_{k=1}^\infty|\xi_k|^p<\infty \right\} ;

2025年3月5日 · On a measure space X, the L^p norm of a function f is |f|_(L^p)=(int_X|f|^p)^(1/p). The L^p-functions are the functions for which this integral converges. For p!=2, the space of L^p-functions is a Banach space which is not a Hilbert space. The L^p-space on R^n, and in most other cases, is the completion of the continuous functions...

Lp(Rn) is the vector space of equivalence classes of integrable functions on Rn, where f is equivalent to g if f = g a.e., such that R 1=p jf jp < 1. We define kf kp = R jf jp : kc f kp = jcj kf kp ; and kf kp = 0 iff f 0. a norm. p q = 1, then fg 2 L1, and kfgk1 kf kpkgkq. Proof. Suffices to consider kf kp = 1 ; kgkq = 1 ; in which case. Proof.

2025年2月11日 · 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 [ 注 1 ] 。 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。

Description: We introduce L^p spaces and the corresponding L^p norms, which are essentially the completion of the normed space of continuous functions under the L^p norm (see the exercises). We then move on to our third unit of the course: Hilbert …

2013年12月28日 · $L^p$ is typically used to indicate $p$-summable functions (with respect to some measure) on a non-discrete measure space, such as the usual $L^p(\mathbb{R})$, the set of functions $f: \mathbb{R} \rightarrow \mathbb{C}$ such …

在分析学中, L p 空间是一类重要的函数空间. 给定测度空间 X 以及实数 p ≥ 1, 函数 f: X → R 的 L p 范数定义为 Lebesgue 积分 ∥ f ∥ p = (∫ X ∣ f (x) ∣ p d x) 1/ p, 所有使得 ∥ f ∥ p 有限的函数 f 构成空间 L p (X). 这一定义也能推广到 p = ∞ 的情况, 以得到空间 L ∞ (X).

2023年11月11日 · In functional analysis, an $L^p$ space is a space of functions for which the $p$-th power of their absolute value is Lebesgue integrable. $L^p$ spaces are sometimes …