实际分析

Mathemagics: A Magical Journey Through Advanced Mathematics Pub Date : 2020-05-18 DOI:10.2307/j.ctt130hk3w.9

Prelim Workshop, Uc Berkeley

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引用次数: 3

摘要

3.由于 Lp 和 Lr 是 CX 的子空间，它们的交集是一个向量空间。显然，‖-‖是一个规范（这直接源于‖-‖p 和‖-‖r 是规范这一事实）。设〈fn〉n=1 是 Lp ∩ Lr 中的考奇序列。由于对于所有 m，n∈ N，‖fm -fn‖p ≤ ‖fm -fn‖，且‖fm -fn‖r ≤ ‖fm -fn‖，显然〈fn〉n=1 在 Lp 和 Lr 中都是考奇序列。设 gp∈Lp 和 gr∈Lr 分别为该序列的极限。给定 ε∈ (0,∞)，存在 N∈ N，对于所有 n∈ N 且 n≥ N，‖fn - gp‖p < ε(p+1)/p。

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Real Analysis

3. Since Lp and Lr are subspaces of CX , their intersection is a vector space. It is clear that ‖ · ‖ is a norm (this follows directly from the fact that ‖ · ‖p and ‖ · ‖r are norms). Let 〈fn〉n=1 be a Cauchy sequence in Lp ∩ Lr. Since ‖fm − fn‖p ≤ ‖fm − fn‖ and ‖fm − fn‖r ≤ ‖fm − fn‖ for all m,n ∈ N, it is clear that 〈fn〉n=1 is a Cauchy sequence in both Lp and Lr. Let gp ∈ Lp and gr ∈ Lr be the respective limits of this sequence. Given ε ∈ (0,∞), there exists N ∈ N such that ‖fn − gp‖p < ε(p+1)/p for all n ∈ N with n ≥ N. If n ∈ N and n ≥ N

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Mathemagics: A Magical Journey Through Advanced Mathematics

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