metric space is a set M together with a distance function ρ(x,y) that represents the distance" between elements a and y of the set M. The distance function must satisfy ) plz,y)20 and plr,y) 0 if and only if y: (ii) p(x,y) o(y,z); (iii) p(z, y) a(x, z) +a(z, y) for all x, y, z in M. Let (M.p) be a metric space. A mapping T from M into M is called a contraction if for some constant a with 0 sa<1, and all a and y in M I. Prove that the real line R becomes a metric space if we define a distance ρ(x, y) between real numbers a and y by Here, your objective is to show that p(z. g) satisfies the three conditions (i), (), and (ii) above. You may assume without proof the "triangle" inequality for the absolute value function for all a, b in R
metric space is a set M together with a distance function ρ(x,y) that represents the distance" between elements a and y of the set M. The distance function must satisfy ) plz,y)20 and plr,y) 0 if and only if y: (ii) p(x,y) o(y,z); (iii) p(z, y) a(x, z) +a(z, y) for all x, y, z in M. Let (M.p) be a metric space. A mapping T from M into M is called a contraction if for some constant a with 0 sa<1, and all a and y in M I. Prove that the real line R becomes a metric space if we define a distance ρ(x, y) between real numbers a and y by Here, your objective is to show that p(z. g) satisfies the three conditions (i), (), and (ii) above. You may assume without proof the "triangle" inequality for the absolute value function for all a, b in R
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:metric space is a set M together with a distance function ρ(x,y) that represents the
distance" between elements a and y of the set M. The distance function must satisfy
) plz,y)20 and plr,y) 0 if and only if y:
(ii) p(x,y) o(y,z);
(iii) p(z, y) a(x, z) +a(z, y) for all x, y, z in M.
Let (M.p) be a metric space. A mapping T from M into M is called a contraction if
for some constant a with 0 sa<1, and all a and y in M
I. Prove that the real line R becomes a metric space if we define a distance ρ(x, y) between
real numbers a and y by
Here, your objective is to show that p(z. g) satisfies the three conditions (i), (), and
(ii) above. You may assume without proof the "triangle" inequality for the absolute
value function
for all a, b in R
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