Definition: Let V be a vector space. A norm is a function v→ ||v|| that satisfies P1: |v|| ≥ 0, for v € V. P2: ||v|| = 0 if and only if v = 0. P3: ||rv|| = |r|||v||, for v EV and r ER. ||u+v|≤|u|| + ||v||, for v € V. P4: -101 The 1-norm and the infinity norm on R² are defined as For example, for 21 = √+ is called the 2-norm on R². = |x1| + |₂| i. Show that the 1-norm satisfies properties P1 to P4. ii. Show that the 1-norm does not satisfy Equation (1). iii. Show that the infinity-norm satisfies properties P1 to P4. iv. Show that the infinity norm does not satisfy Equation (1). (That is, we cannot write ||x||₁1 nor ||x|| as √(x,x), for any inner product on R².) and = = max(₁,₂), respectively.

Definition: Let V be a vector space. A norm is a function v→ ||v|| that satisfies P1: |v|| ≥ 0, for v € V. P2: ||v|| = 0 if and only if v = 0. P3: ||rv|| = |r|||v||, for v EV and r ER. ||u+v|≤|u|| + ||v||, for v € V. P4: -101 The 1-norm and the infinity norm on R² are defined as For example, for 21 = √+ is called the 2-norm on R². = |x1| + |₂| i. Show that the 1-norm satisfies properties P1 to P4. ii. Show that the 1-norm does not satisfy Equation (1). iii. Show that the infinity-norm satisfies properties P1 to P4. iv. Show that the infinity norm does not satisfy Equation (1). (That is, we cannot write ||x||₁1 nor ||x|| as √(x,x), for any inner product on R².) and = = max(₁,₂), respectively.

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

Definition: Let V be a vector space. A norm is a function v→ ||v|| that satisfies
P1: |v|| ≥ 0, for v € V.
P2:
||v|| = 0 if and only if v = 0.
P3:
||rv|| = |r|||v||, for v EV and r ER.
||u+v|≤|u|| + ||v||, for v € V.
P4:
-101
The 1-norm and the infinity norm on R² are defined as
For example, for
21
= √+ is called the 2-norm on R².
= |x1| + |₂|
i. Show that the 1-norm satisfies properties P1 to P4.
ii. Show that the 1-norm does not satisfy Equation (1).
iii. Show that the infinity-norm satisfies properties P1 to P4.
iv. Show that the infinity norm does not satisfy Equation (1).
(That is, we cannot write ||x||₁1 nor ||x|| as √(x,x), for any inner product on R².)
and
=
= max(₁,₂), respectively.

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