Answered: ∥x∥2 ≤ ∥x∥1 ≤ √n∥x∥2,

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∥x∥2 ≤ ∥x∥1 ≤ √n∥x∥2,

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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Question

∥.∥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

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Step 1: Application of Cauchy schwarz inequality.

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Step 2: Proving norms are equivalent.

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