A metric space is a set M together with a distance function ρ(x,y) that represents between elements and y of the set M. The distance function must satisfy the (i) ρ(z, y) > 0 and ρ(x,y)-0 if and only if x y; (iii) ρ(z, y) a(x, z) + ρ(z, y) for all x, y, s in M. Let (M. p) be a metric space. A mapping T from M into M is called a contraction if p(Tr, Ty) ap(x, y) for some constant α with 0 a < 1, and all 21.and y in M. 2. For points P (, vi) and P(2/2)i the plane R2 define the "taxicab" metric which is the distance between two points in a city if one travels only along rectangularly- aligned city streets. Prove that (R, ρ) is a metric space by showing that (i), (ii), and (iii) are again satisfied
A metric space is a set M together with a distance function ρ(x,y) that represents between elements and y of the set M. The distance function must satisfy the (i) ρ(z, y) > 0 and ρ(x,y)-0 if and only if x y; (iii) ρ(z, y) a(x, z) + ρ(z, y) for all x, y, s in M. Let (M. p) be a metric space. A mapping T from M into M is called a contraction if p(Tr, Ty) ap(x, y) for some constant α with 0 a < 1, and all 21.and y in M. 2. For points P (, vi) and P(2/2)i the plane R2 define the "taxicab" metric which is the distance between two points in a city if one travels only along rectangularly- aligned city streets. Prove that (R, ρ) is a metric space by showing that (i), (ii), and (iii) are again satisfied
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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Transcribed Image Text:A metric space is a set M together with a distance function ρ(x,y) that represents
between elements and y of the set M. The distance function must satisfy
the
(i) ρ(z, y) > 0 and ρ(x,y)-0 if and only if x
y;
(iii) ρ(z, y) a(x, z) + ρ(z, y) for all x, y, s in M.
Let (M. p) be a metric space. A mapping T from M into M is called a contraction if
p(Tr, Ty)
ap(x, y)
for some constant α with 0
a < 1, and all 21.and y in M.
2. For points P
(, vi) and P(2/2)i the plane R2 define the "taxicab" metric
which is the distance between two points in a city if one travels only along rectangularly-
aligned city streets. Prove that (R, ρ) is a metric space by showing that (i), (ii), and
(iii) are again satisfied
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