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1.9. Consider the space R² with the norm ||- ||p, introduced in Section [1.5 For p=1,2,∞o draw the "unit ball" B, in the norm ||- || B₂ := {x € R² : ||x|| p ≤ 1}. Can you guess what the balls B₂ for other p look like?

1.9. Consider the space R² with the norm ||- ||p, introduced in Section [1.5 For p=1,2,∞o draw the "unit ball" B, in the norm ||- || B₂ := {x € R² : ||x|| p ≤ 1}. Can you guess what the balls B₂ for other p look like?

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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1.9. Consider the space R2 with the norm ||- ||p, introduced in Section [1.5 For p = 1,2,00 draw the "unit ball" B, in the norm || ||p B₂ := {x € R² : ||x||p ≤ 1}. Can you guess what the balls Bp for other p look like?
Transcribed Image Text:1.9. Consider the space R2 with the norm ||- ||p, introduced in Section [1.5 For p = 1,2,00 draw the "unit ball" B, in the norm || ||p B₂ := {x € R² : ||x||p ≤ 1}. Can you guess what the balls Bp for other p look like?
1.5. Norm. Normed spaces. We have proved before that the norm ||v|| satisfies the following properties: 1. Homogeneity: ||av|| = |a|-||v|| for all vectors v and all scalars a. 2. Triangle inequality: ||u+v|| ≤ ||u|| + ||v||. 3. Non-negativity: ||v|| ≥0 for all vectors v. 4. Non-degeneracy: ||v|| = 0 if and only if v = 0. Suppose in a vector space V we assigned to each vector v a number ||v|| such that above properties 1-4 are satisfied. Then we say that the function v → ||v|| is a norm. A vector space V equipped with a norm is called a normed space.
Transcribed Image Text:1.5. Norm. Normed spaces. We have proved before that the norm ||v|| satisfies the following properties: 1. Homogeneity: ||av|| = |a|-||v|| for all vectors v and all scalars a. 2. Triangle inequality: ||u+v|| ≤ ||u|| + ||v||. 3. Non-negativity: ||v|| ≥0 for all vectors v. 4. Non-degeneracy: ||v|| = 0 if and only if v = 0. Suppose in a vector space V we assigned to each vector v a number ||v|| such that above properties 1-4 are satisfied. Then we say that the function v → ||v|| is a norm. A vector space V equipped with a norm is called a normed space.
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