(a) Consider the metric space < X,o for the case in which the metric o is the discrete metric 0 if x = y o(x, y) = { 1 if x#y Find the closed ball B,(a) c X with centre a E X and radius r >0 (i) if r < 1, (ii) if r > 1.

(a) Consider the metric space < X,o for the case in which the metric o is the discrete metric 0 if x = y o(x, y) = { 1 if x#y Find the closed ball B,(a) c X with centre a E X and radius r >0 (i) if r < 1, (ii) if r > 1.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...

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Question

Given the set X=R^4 of points in 4-D Euclidean space,required to answer follow up qu

idean
(a) Consider the metric space < X,o > for the case in which the metric ơ is the discrete
metric
S0 if
1 if x #y
x = Y
o (x, y)
Find the closed ball B,(a) c X with centre a E X and radius r > 0
(i) if r < 1,
(ii) if r > 1.
(b) Next, consider the metric space < X, d > for the case in which the metric d is the usual
metric on R'. Given the closed ball B,(a) C X with centre a =
P(3, 1, 1, 1) that is located on its boundary OB.
(2,0, 2, 2) and the point
(i) Show that every point x 4 Br(a) is the centre of an open ball B:(x) with some
feasible radius e > 0, and give the feasible range for ɛ.
(ii) Use this to prove that the complement B„(a)° of the close ball is an open set.

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consider the metric space < X, d > for the case in which the metric d is the usual metric on R'. Given the closed ball B,(a) C X with centre a = P(3, 1, 1, 1) that is located on its boundary OB. (2,0, 2, 2) and the point (i) Show that every point x 4 Br(a) is the centre of an open ball B:(x) with some feasible radius e > 0, and give the feasible range for ɛ. (ii) Use this to prove that the complement B„(a)° of the close ball is an open set.

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