Exercise 4. Suppose K is a compact subset of the metric space (X, d). a) Show that for each x X, there is a point k such that d(x, kx) = dk(x) = d(x, K) = info d(k, x). b) Endow R² with the metric dmax. If K = [−1, 1] × {0} C R², find a point (x, y) = R² such that dmax ((x, y), K) = dmax ((x, y), (t,0)), for all t € [−1,1]. c) Endow R² with the Euclidean metric metric d₂. If K = [−1, 1] × {0} ℃ R², show that for every (x, y) = R² there is a unique t € [-1,1] with d₂((x, y), K) = d₂((x, y), (t,0)).
Exercise 4. Suppose K is a compact subset of the metric space (X, d). a) Show that for each x X, there is a point k such that d(x, kx) = dk(x) = d(x, K) = info d(k, x). b) Endow R² with the metric dmax. If K = [−1, 1] × {0} C R², find a point (x, y) = R² such that dmax ((x, y), K) = dmax ((x, y), (t,0)), for all t € [−1,1]. c) Endow R² with the Euclidean metric metric d₂. If K = [−1, 1] × {0} ℃ R², show that for every (x, y) = R² there is a unique t € [-1,1] with d₂((x, y), K) = d₂((x, y), (t,0)).
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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Question
4
Need a b and c
![Exercise 4. Suppose K is a compact subset of the metric space (X, d).
a) Show that for each x E X, there is a point ke such that
d(x, kr) = dk (x) = d(x, K) = inf d(k, x).
KEK
b) Endow R2 with the metric dmax. If K = [−1, 1] × {0} CR², find a point (x, y) = R² such that
dmax ((x, y), K) = dmax ((x, y), (t,0)), for all t € [1,1].
c) Endow R2 with the Euclidean metric metric d₂. If K = [−1, 1] × {0} C R², show that for every
(x, y) = R² there is a unique t € [-1,1] with
d₂((x, y), K) = d₂((x, y), (t,0)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2Ff913d5fc-1a31-40ad-82c6-2cc7163fcdf5%2F1ilh1t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 4. Suppose K is a compact subset of the metric space (X, d).
a) Show that for each x E X, there is a point ke such that
d(x, kr) = dk (x) = d(x, K) = inf d(k, x).
KEK
b) Endow R2 with the metric dmax. If K = [−1, 1] × {0} CR², find a point (x, y) = R² such that
dmax ((x, y), K) = dmax ((x, y), (t,0)), for all t € [1,1].
c) Endow R2 with the Euclidean metric metric d₂. If K = [−1, 1] × {0} C R², show that for every
(x, y) = R² there is a unique t € [-1,1] with
d₂((x, y), K) = d₂((x, y), (t,0)).
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