(a) Let 0 < p s co and X = C[0, 1] be the space of continuous real valued functions on the interval [0, 1]. Consider X with 1/p |x(1)" for 0 < p< 00 and any x e C[0, 1] %3! = max |x(1)), for p = 0 and any x e C[0, 1]. re(0,1] i. For which p, is X a normed space? For which p is X a Banach space? (No proof is needed.) ii. Show that the triangle inequality is not valid for 0 < p < 1.
(a) Let 0 < p s co and X = C[0, 1] be the space of continuous real valued functions on the interval [0, 1]. Consider X with 1/p |x(1)" for 0 < p< 00 and any x e C[0, 1] %3! = max |x(1)), for p = 0 and any x e C[0, 1]. re(0,1] i. For which p, is X a normed space? For which p is X a Banach space? (No proof is needed.) ii. Show that the triangle inequality is not valid for 0 < p < 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
![(a) Let 0 < p s co and X = C[0, 1] be the space of continuous real valued functions on
the interval [0, 1]. Consider X with
1/p
|x(1)"
for 0 < p< 00 and any x e C[0, 1]
%3!
= max |x(1)), for p = 0 and any x e C[0, 1].
re(0,1]
i. For which p, is X a normed space? For which p is X a Banach space? (No proof
is needed.)
ii. Show that the triangle inequality is not valid for 0 < p < 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad4d02b6-78f1-4c0a-a713-3b04a8400050%2F50213879-af57-4cdb-a49c-212ab5cf9fde%2Fpoe16fm_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let 0 < p s co and X = C[0, 1] be the space of continuous real valued functions on
the interval [0, 1]. Consider X with
1/p
|x(1)"
for 0 < p< 00 and any x e C[0, 1]
%3!
= max |x(1)), for p = 0 and any x e C[0, 1].
re(0,1]
i. For which p, is X a normed space? For which p is X a Banach space? (No proof
is needed.)
ii. Show that the triangle inequality is not valid for 0 < p < 1.
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