1. Show that there exists a linear continuous form on L ([0, 1]) such that o(f) = f(0) if f is continuous. 2. Using Riemann-Lebesgue Lemma, show that there is not exists any g € L¹([0, 1]) such that o(f) f f(t)g(t)dt for all f € L% ([0, 1]). 3. What you can deduce from the previous parts.
1. Show that there exists a linear continuous form on L ([0, 1]) such that o(f) = f(0) if f is continuous. 2. Using Riemann-Lebesgue Lemma, show that there is not exists any g € L¹([0, 1]) such that o(f) f f(t)g(t)dt for all f € L% ([0, 1]). 3. What you can deduce from the previous parts.
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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![Problem 4.
1. Show that there exists a linear continuous form on L ([0, 1]) such that o(f) = f(0)
if f is continuous.
2. Using Riemann-Lebesgue Lemma, show that there is not exists any g € L¹([0, 1])
such that o(f) = f f(t)g(t)dt for all f = L*([0, 1]).
3. What you can deduce from the previous parts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9612a6a5-e19c-4346-ae6e-6941d4a9f0e7%2F2934f32e-b36c-4cae-87f5-351eb16fb9f8%2F9w8lzp9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4.
1. Show that there exists a linear continuous form on L ([0, 1]) such that o(f) = f(0)
if f is continuous.
2. Using Riemann-Lebesgue Lemma, show that there is not exists any g € L¹([0, 1])
such that o(f) = f f(t)g(t)dt for all f = L*([0, 1]).
3. What you can deduce from the previous parts.
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