#3 Need parts 1 and 2 Exercise 3. 1) Consider an normed vector space (E, |-||). As usual themetric d on E is d(x, y) = ||yx||. Show that the closure of the open ballB(x,r) is the closed ball B(x,r), for any x ER" and any r > 0. Whathappens when r = 0?2) Find a metric space (X, d) for which we can find an open ball Å(x,r),for some x € X and r > 0, whose closure is not B(x,r).

Answered: Image /qna-images/answer/8c9a7dbf-91f2-4838-974b-38211b8e3c4c.jpg

Exercise 3. 1) Consider an normed vector space (E, ||-||). As usual the metric d on E is d(x, y) = ||yx||. Show that the closure of the open ball B(x,r) is the closed ball B(x, r), for any x € R" and any r > 0. What happens when r = 0? 2) Find a metric space (X, d) for which we can find an open ball B(x,r), for some x EX and r > 0, whose closure is not B(x,r).

Exercise 3. 1) Consider an normed vector space (E, ||-||). As usual the metric d on E is d(x, y) = ||yx||. Show that the closure of the open ball B(x,r) is the closed ball B(x, r), for any x € R" and any r > 0. What happens when r = 0? 2) Find a metric space (X, d) for which we can find an open ball B(x,r), for some x EX and r > 0, whose closure is not B(x,r).

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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