Question 2. functions f : [0, 1] → R. Define an integral norm || · ||i by Let C0, 1] represent the collection of continuous |f(x)| dx. We then introduce a distance function p which is given in terms of the integral norm || · ||i as P(f,g) := ||f – 9g||i, Vf,g € C[0, 1] To show that < C[0, 1], p > is a metric space: (a) Prove that the distance function p positive definite on C[0, 1]. (b) Is the distance function p is symmetric? Justify. (c) Show that p satisfies the triangle inequality on C[0, 1].
Question 2. functions f : [0, 1] → R. Define an integral norm || · ||i by Let C0, 1] represent the collection of continuous |f(x)| dx. We then introduce a distance function p which is given in terms of the integral norm || · ||i as P(f,g) := ||f – 9g||i, Vf,g € C[0, 1] To show that < C[0, 1], p > is a metric space: (a) Prove that the distance function p positive definite on C[0, 1]. (b) Is the distance function p is symmetric? Justify. (c) Show that p satisfies the triangle inequality on C[0, 1].
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 23E
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![Let C0, 1] represent the collection of continuous
Question 2.
functions f : [0, 1] → R. Define an integral norm || · ||i by
||/ ||; := / \f(x)| dar.
We then introduce a distance function p which is given in terms of the integral norm || · |li as
P(f, 9) := ||f – 9g||i, Vf, g € C[0, 1]
To show that < C[0, 1], p > is a metric space:
(a) Prove that the distance function p positive definite on C[0, 1].
(b) Is the distance function p is symmetric? Justify.
(c) Show that p satisfies the triangle inequality on C[0, 1].
(d) Conclude whether or not < C[0, 1], p > is a metric space. Would you say that it is a
complete metric space or not? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6d34efb-714e-4109-86a1-c1e5ba7c23ed%2Fa76aad3e-630a-4ef3-841a-9d3a01db763e%2Fr96wa8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let C0, 1] represent the collection of continuous
Question 2.
functions f : [0, 1] → R. Define an integral norm || · ||i by
||/ ||; := / \f(x)| dar.
We then introduce a distance function p which is given in terms of the integral norm || · |li as
P(f, 9) := ||f – 9g||i, Vf, g € C[0, 1]
To show that < C[0, 1], p > is a metric space:
(a) Prove that the distance function p positive definite on C[0, 1].
(b) Is the distance function p is symmetric? Justify.
(c) Show that p satisfies the triangle inequality on C[0, 1].
(d) Conclude whether or not < C[0, 1], p > is a metric space. Would you say that it is a
complete metric space or not? Justify your answer.
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