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∥x∥2 ≤ ∥x∥1 ≤ √n∥x∥2,

∥x∥2 ≤ ∥x∥1 ≤ √n∥x∥2,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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∥.∥1, ∥.∥2, and∥.∥∞ are norms on R.
Show the following relations for x∈R and thereby the called equivalence of norms:

a)  ∥x∥2 ≤ ∥x∥1 ≤ √n∥x∥2,

I understand the first part of the solution(picture) that 

∥x∥2 ≤ ∥x∥1. I am greatly struggeling to understand the solution of :  ∥x∥1 ≤ √n∥x∥2.

Can somebody explain the solution to me?

a.) Es gilt: da alle Mischterme auf der rechten ||||2 = = √√√x² + + x 2²/2 ≤ √(|x₁| + ··· +|xn|)² = |x1| + ··· + | Xxn| = ||*||1, · x ² + + x²³² ≤ (|x₁| +...+|xn|) ² Seite positiv sind. Dies zeigt mithilfe der Wurzel: also ||||2||||1₁. Auf der anderen Seite gilt: |||||₁=(x₁,, |xn|) und somit ||||1 ≤ √√n||x||2. 3 (|ux|'...'|¹x|) = OFIQUIOL- 2 = √n||x||2
Transcribed Image Text:a.) Es gilt: da alle Mischterme auf der rechten ||||2 = = √√√x² + + x 2²/2 ≤ √(|x₁| + ··· +|xn|)² = |x1| + ··· + | Xxn| = ||*||1, · x ² + + x²³² ≤ (|x₁| +...+|xn|) ² Seite positiv sind. Dies zeigt mithilfe der Wurzel: also ||||2||||1₁. Auf der anderen Seite gilt: |||||₁=(x₁,, |xn|) und somit ||||1 ≤ √√n||x||2. 3 (|ux|'...'|¹x|) = OFIQUIOL- 2 = √n||x||2
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