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Exercise 6. Consider an normed vector space (E, ||-||). As usual the metric d on E is d(x, y) = ||yx||. a) Show that the sphere Sa(x,r) = {y ≤ E | d(x, y) = r} = has an empty interior, for every x EE and every finite r≥ 0. b) For any r> 0 and x = E, show that Int (Ba(x,r)), the interior of the closed ball Ba(x,r) is the open ball Ba(x, r).

Exercise 6. Consider an normed vector space (E, ||-||). As usual the metric d on E is d(x, y) = ||yx||. a) Show that the sphere Sa(x,r) = {y ≤ E | d(x, y) = r} = has an empty interior, for every x EE and every finite r≥ 0. b) For any r> 0 and x = E, show that Int (Ba(x,r)), the interior of the closed ball Ba(x,r) is the open ball Ba(x, r).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.2: Inner Product Spaces
Problem 101E: Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that...
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Exercise 6. Consider an normed vector space (E, ||·||). As usual the metric d on E is d(x, y) = ||y — x||. a) Show that the sphere Sa(x,r) {y € E | d(x, y) = r} = has an empty interior, for every x EE and every finite r≥ 0. b) For any r> 0 and x € E, show that Int(Ba(x,r)), the interior of the closed ball B₁(x,r) is the open ball B₁(x, r). =
Transcribed Image Text:Exercise 6. Consider an normed vector space (E, ||·||). As usual the metric d on E is d(x, y) = ||y — x||. a) Show that the sphere Sa(x,r) {y € E | d(x, y) = r} = has an empty interior, for every x EE and every finite r≥ 0. b) For any r> 0 and x € E, show that Int(Ba(x,r)), the interior of the closed ball B₁(x,r) is the open ball B₁(x, r). =
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