Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as8(M)= sup d(x, y)хуем(≤00).Show that 8 (M)= 0 if and only if M contains only one point.

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Answered: Let (X, d) be a metric space. The…

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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Q3/ Choose the correct answer a, b, c, or d. 1) The compactness of a subset V of a normed space F is (a) Sufficient and necessary condition (b) Necessary condition (e) Sufficient condition (d) all of them true, to existences of the best approximation. 2) Let F = R? and V ((x,y): (x- 2) + y? 2 9), for f = (2,0) and || | = II , then the best approximation for f from V is (a) x + y? 3 (b) (x- 2) + y? 9 (c) (x- 1)2 +y? 3 (d) (x - 1)2 + y?> 9. 3) If E has a strictly convex norm, then for any subspace B of E and any points ge E, there exists (a) one best approximation (b) two best approximation (c) infinity best approximation (d) no answer. 4) A subset A of a vector space V is said to be convex if for any x,y E A, we have ux+ (1- )y E A, for each (a) 0 Sus1 (b) 0 0 (P) I>n50 (0) 5) Every normed space is (a) metric space (b) vector space (c) a and b (d) not any one of them.
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Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x, y) (≤00). хуем Show that 8(M)= 0 if and only if M contains only one point.