3. Let (X, d) bea metricandP: X→IRspace,desy)(+ doxy)de fined by uM pusy) =show that pP is a metric.

Answered: Let (X, d) be a metric space, and p: X→…

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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3. Let (X, d) be
a metric
and
P: X→IR
space,
desy)
(+ doxy)
de fined by uM pusy) =
show that p
P is a metric.

Expert Solution

Step 1

Consider a set X and a function defined from it to the real numbers as $d : X \times X \to ℝ$ . Then, $(X, d)$ is a metric space if it satisfies the below conditions.

For all $x, y \in X$ , $d (x, y) \geq 0$ .
The function $d (x, y) = 0$ only if $x = y$ .
For all $x, y \in X$ , $d (x, y) = d (y, x)$ .
For all $x, y, z \in X$ , $d (x, y) \leq d (x, z) + d (z, y)$ .

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