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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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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