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Problem 6.1.* A function N: RnR is called a norm on Rn if it has the following properties for all x, y E R" and a ER: (i) N(x) > 0. (ii) N(x) = 0 if and only if x = 0. (iii) N(ax) = |a|N(x). (iv) N(x+y) ≤ N(x) + N(y). Prove that there exists an M> 0 such that Prove also that N(x) ≤ Mx for all x E R. IN(x) - N(y)| ≤ Mx-y|| for all x, y ER".

Problem 6.1.* A function N: RnR is called a norm on Rn if it has the following properties for all x, y E R" and a ER: (i) N(x) > 0. (ii) N(x) = 0 if and only if x = 0. (iii) N(ax) = |a|N(x). (iv) N(x+y) ≤ N(x) + N(y). Prove that there exists an M> 0 such that Prove also that N(x) ≤ Mx for all x E R. IN(x) - N(y)| ≤ Mx-y|| for all x, y ER".

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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Problem 6.1. A function N: R" → R is called a norm on Rn if it has the following properties for all x, y E R" and a ER: (i) N(x) > 0. (ii) N(x) = 0 if and only if x = 0. (iii) N(ax) = |a|N(x). (iv) N(x + y) ≤ N(x) + N(y). Prove that there exists an M> 0 such that Prove also that N(x) ≤ Mx for all x ER". |N(x) = N(y)| ≤ Mx-y|| for all x, y € R". CDn we define
Transcribed Image Text:Problem 6.1. A function N: R" → R is called a norm on Rn if it has the following properties for all x, y E R" and a ER: (i) N(x) > 0. (ii) N(x) = 0 if and only if x = 0. (iii) N(ax) = |a|N(x). (iv) N(x + y) ≤ N(x) + N(y). Prove that there exists an M> 0 such that Prove also that N(x) ≤ Mx for all x ER". |N(x) = N(y)| ≤ Mx-y|| for all x, y € R". CDn we define
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