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Exercise 4 (The max norm ||· ||). Recall from class that the max norm ||· ||∞ is defined on Rn as ||x||∞ : = = max{|x₁,..., |xn|}. Show that | || is a norm on R. More precisely, show that the following hold. : 0 holds if and only if x is the zero (i) |||| ≥0 for all x € Rn and ||x|| = ∞ vector. (ii) ||ax||∞ = |a|||x||∞ for all a € R and for all x € R”. (iii) ||x+y||∞ ≤ ||x||∞ + ||y||∞ for all x, y ≤ Rn.

Exercise 4 (The max norm ||· ||). Recall from class that the max norm ||· ||∞ is defined on Rn as ||x||∞ : = = max{|x₁,..., |xn|}. Show that | || is a norm on R. More precisely, show that the following hold. : 0 holds if and only if x is the zero (i) |||| ≥0 for all x € Rn and ||x|| = ∞ vector. (ii) ||ax||∞ = |a|||x||∞ for all a € R and for all x € R”. (iii) ||x+y||∞ ≤ ||x||∞ + ||y||∞ for all x, y ≤ Rn.

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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Exercise 4 (The max norm ||||). Recall from class that the max norm ||· || is defined on Rn as = = max{|x₁|, ...,|xn|}. ||x||∞ : Show that | || is a norm on R". More precisely, show that the following hold. (i) |||| ≥0 for all x € Rn and ||x|| 0 holds if and only if x is the zero = ∞ vector. (ii) ||ax||∞ = |a|||x||∞ for all a € R and for all x € R”. (iii) ||x+y||∞ ≤ ||x||∞ + ||y||∞ for all x, y = R".
Transcribed Image Text:Exercise 4 (The max norm ||||). Recall from class that the max norm ||· || is defined on Rn as = = max{|x₁|, ...,|xn|}. ||x||∞ : Show that | || is a norm on R". More precisely, show that the following hold. (i) |||| ≥0 for all x € Rn and ||x|| 0 holds if and only if x is the zero = ∞ vector. (ii) ||ax||∞ = |a|||x||∞ for all a € R and for all x € R”. (iii) ||x+y||∞ ≤ ||x||∞ + ||y||∞ for all x, y = R".
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