Proof: ExerciseExercises(1.) Prove that (R", d) with d : R" x R" → R defined byd(r, y) = >(*k – Yk)²k=1is a metric space. [x = (x1, 2,..., xn) and y = (yı, Y2, ..., Yn) are elements of R7-](2.) Let (S, d) be a metric space and suppose that p: S x S → R is defined byd(x, y)1+d(x,y)P(x, y) =for all points r, y E S. Prove that (S, p) is a metric space, that it is bounded and thatp(x, y) < d(x, y) for all r, y E S. [Hint: For the triangle inequality you may use f(t) = andf'(t) = > 0 for all t, i,e., f is an increasing function.]12:42 AMP Type here to searchヘ口 )5/12/2021

Since you have asked multiple question, we will solve the first question for you. If youwant any…

Answered: (1.) Prove that (R", d) with d : R" ×…

(1.) Prove that (R", d) with d : R" × R" → R defined by d(r, y) = E(Ik – yk)² k=1 is a metric space. [x = (x1,#2,..,In) and y = (y1, 42; -... Yn) are elements of R"-] %3D

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.2: The Kernewl And Range Of A Linear Transformation

Problem 69E

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Question

Proof: Exercise
Exercises
(1.) Prove that (R", d) with d : R" x R" → R defined by
d(r, y) = >(*k – Yk)²
k=1
is a metric space. [x = (x1, 2,..., xn) and y = (yı, Y2, ..., Yn) are elements of R7-]
(2.) Let (S, d) be a metric space and suppose that p: S x S → R is defined by
d(x, y)
1+d(x,y)
P(x, y) =
for all points r, y E S. Prove that (S, p) is a metric space, that it is bounded and that
p(x, y) < d(x, y) for all r, y E S. [Hint: For the triangle inequality you may use f(t) = and
f'(t) = > 0 for all t, i,e., f is an increasing function.]
12:42 AM
P Type here to search
ヘ口 )
5/12/2021

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