Let X = C[0,4], with the usual notation and suppose X is endowed with the norm, ||f||2 Consider the sequence {9} defined by n=1 9n(t) = = 1, = (t-2), Verify that (i) 9n C[0, 4] for each n; (ii) {9} is a Cauchy sequence; n=1 (iii) gn →g as n→∞, where (L'iscopar)¹ g(t) 0, if 0 ≤t≤ 2; if 2 < t < 4. 1, Conclude that C[0, 4] with ||.||2 norm is not a complete space. = if 0 ≤ t ≤ 2; if 2 ≤ t ≤ 2 + ²/3; if 2 + / ≤ t ≤ 4.
Let X = C[0,4], with the usual notation and suppose X is endowed with the norm, ||f||2 Consider the sequence {9} defined by n=1 9n(t) = = 1, = (t-2), Verify that (i) 9n C[0, 4] for each n; (ii) {9} is a Cauchy sequence; n=1 (iii) gn →g as n→∞, where (L'iscopar)¹ g(t) 0, if 0 ≤t≤ 2; if 2 < t < 4. 1, Conclude that C[0, 4] with ||.||2 norm is not a complete space. = if 0 ≤ t ≤ 2; if 2 ≤ t ≤ 2 + ²/3; if 2 + / ≤ t ≤ 4.
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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![(2) Let X = C[0,4], with the usual notation and suppose X is endowed with the
norm,
||||2 = ([^\f(t)1³dt) ³
Consider the sequence {9}1 defined by
n=1
0,
gn(t) =
(t-2),
1,
Verify that
(i) 9n C[0, 4] for each n;
(ii) {9} is a Cauchy sequence;
(iii) gng as n → ∞, where
n=1
if 0 ≤ t ≤ 2;
if 2 ≤t≤ 2+
if 2 + ² ≤ t ≤ 4.
0,
if 0 ≤ t ≤ 2;
1,
if 2 < t < 4.
Conclude that C[0, 4] with ||.||2 norm is not a complete space.
g(t) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d0f188d-b9e0-49c3-bbf4-5bef0fd4e153%2F41f4b25d-5e6b-4f9d-bc7d-b757e4a9f410%2Fopdk40j_processed.png&w=3840&q=75)
Transcribed Image Text:(2) Let X = C[0,4], with the usual notation and suppose X is endowed with the
norm,
||||2 = ([^\f(t)1³dt) ³
Consider the sequence {9}1 defined by
n=1
0,
gn(t) =
(t-2),
1,
Verify that
(i) 9n C[0, 4] for each n;
(ii) {9} is a Cauchy sequence;
(iii) gng as n → ∞, where
n=1
if 0 ≤ t ≤ 2;
if 2 ≤t≤ 2+
if 2 + ² ≤ t ≤ 4.
0,
if 0 ≤ t ≤ 2;
1,
if 2 < t < 4.
Conclude that C[0, 4] with ||.||2 norm is not a complete space.
g(t) =
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