Problem 2. Let C([0, 1]) be the set of real continuous functions on [0, 1], induced with the norm ||.||, and let X = C¹([0, 1]) be a closed vector subspace of C([0, 1]), i.e.every element of X is continuously differentiable. Define T: X→ C([0, 1]) by T(f) = f'. 1. Show that the graph of T is closed. 2. Deduce that there exists a positive integer N such that f'||< N for all f EX such that ||f|| < 1. 3. Set In = for all 0 ≤ n < N and define S: X → RN+¹ by S(ƒ) = (ƒ (xo), f(x₁),..., f(xN)). (a) Suppose that ||f|| = 1 and S(f) = 0. Show, using the Mean Value Theorem, that leads to a contradiction. (b) Deduce that X has finite dimension and dim X < N + 1.
Problem 2. Let C([0, 1]) be the set of real continuous functions on [0, 1], induced with the norm ||.||, and let X = C¹([0, 1]) be a closed vector subspace of C([0, 1]), i.e.every element of X is continuously differentiable. Define T: X→ C([0, 1]) by T(f) = f'. 1. Show that the graph of T is closed. 2. Deduce that there exists a positive integer N such that f'||< N for all f EX such that ||f|| < 1. 3. Set In = for all 0 ≤ n < N and define S: X → RN+¹ by S(ƒ) = (ƒ (xo), f(x₁),..., f(xN)). (a) Suppose that ||f|| = 1 and S(f) = 0. Show, using the Mean Value Theorem, that leads to a contradiction. (b) Deduce that X has finite dimension and dim X < N + 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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Please solve the following exercise
![Problem 2. Let C([0, 1]) be the set of real continuous functions on [0, 1], induced with
the norm ||.||, and let X = C¹([0, 1]) be a closed vector subspace of C([0, 1]), i.e.every
element of X is continuously differentiable. Define T: X → C([0, 1]) by T(ƒ) = f'.
1. Show that the graph of T is closed.
2. Deduce that there exists a positive integer N such that f'||< N for all fe X
such that ||f|| < 1.
3. Set In
for all 0 ≤ n ≤ N and define S : X → RN+¹ by S(ƒ) = (ƒ (xo), ƒ(x₁), …..‚ ƒ(xN)).
(a) Suppose that ||f|| = 1 and S(f) = 0. Show, using the Mean Value Theorem,
that leads to a contradiction.
(b) Deduce that X has finite dimension and dim X < N + 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9612a6a5-e19c-4346-ae6e-6941d4a9f0e7%2F0088a977-6a70-4145-bc68-62a4ff5b543f%2F8ox6jj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2. Let C([0, 1]) be the set of real continuous functions on [0, 1], induced with
the norm ||.||, and let X = C¹([0, 1]) be a closed vector subspace of C([0, 1]), i.e.every
element of X is continuously differentiable. Define T: X → C([0, 1]) by T(ƒ) = f'.
1. Show that the graph of T is closed.
2. Deduce that there exists a positive integer N such that f'||< N for all fe X
such that ||f|| < 1.
3. Set In
for all 0 ≤ n ≤ N and define S : X → RN+¹ by S(ƒ) = (ƒ (xo), ƒ(x₁), …..‚ ƒ(xN)).
(a) Suppose that ||f|| = 1 and S(f) = 0. Show, using the Mean Value Theorem,
that leads to a contradiction.
(b) Deduce that X has finite dimension and dim X < N + 1.
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