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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
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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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 ER" and any r > 0. What happens 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).
Transcribed Image Text: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 ER" and any r > 0. What happens 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).
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