Question 2. functions f : [0, 1] → R. Define an integral norm || · ||i by Let C[0, 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,9) := || f – 9|li Vf, g € C[0, 1] To show that < C0, 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 C[0, 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,9) := || f – 9|li Vf, g € C[0, 1] To show that < C0, 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].
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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![Question 2.
functions f : [0, 1] → R. Define an integral norm || - ||i by
Let C[0, 1] represent the collection of continuous
||f||: := / \f(x)| dx.
0.
We then introduce a distance function p which is given in terms of the integral norm || · Ili as
P(f,9) := ||f – 9|li, 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%2Fd74c38e2-bc89-4af1-ad7f-05db36328edb%2Fc07f42a4-e93f-4182-83a6-a188c754c0ee%2Fci5vfj6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 2.
functions f : [0, 1] → R. Define an integral norm || - ||i by
Let C[0, 1] represent the collection of continuous
||f||: := / \f(x)| dx.
0.
We then introduce a distance function p which is given in terms of the integral norm || · Ili as
P(f,9) := ||f – 9|li, 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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