sup norm
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![2. Let
A = {ƒ\ƒ € C° ([0, 1]), ||f||sup ≤ 1}
where the sup norm is taken over [0, 1].
(a) Show that under the sup norm, ✅ is closed and bounded.
(b) Let
fn(2) :=
0,
2n+2x2,
- 2n+2x + 4,
0,
i. Show that for n € N, fn € A.
ii. Find
for n‡m.
iii. Show that A is not compact.
if x = [0, 2-n-1];
if x € [2-n-¹, 2+2];
3
3
E
if x = [2²³+2, 2-"];
if x = [2-", 1]
||fn - fm|| sup](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2Fa1b59c88-273f-49f3-9512-642806a03a21%2F7k76v84_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let
A = {ƒ\ƒ € C° ([0, 1]), ||f||sup ≤ 1}
where the sup norm is taken over [0, 1].
(a) Show that under the sup norm, ✅ is closed and bounded.
(b) Let
fn(2) :=
0,
2n+2x2,
- 2n+2x + 4,
0,
i. Show that for n € N, fn € A.
ii. Find
for n‡m.
iii. Show that A is not compact.
if x = [0, 2-n-1];
if x € [2-n-¹, 2+2];
3
3
E
if x = [2²³+2, 2-"];
if x = [2-", 1]
||fn - fm|| sup
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