1. Show that the graph of T is closed. 2. Deduce that there exists a positive integer N such that f'oN for all f € X such that f. 1.
1. Show that the graph of T is closed. 2. Deduce that there exists a positive integer N such that f'oN for all f € X such that f. 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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part 1 2 lineare operator close graph theoream
![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 € X
such that f < 1.
3. Set En for all 0 ≤ n ≤ N and define S: X → RN+¹ by S(f) = (f(xo), f(x₁),..., ƒ(EN)).
=
(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%2F4845eae0-c46b-40ae-8f6b-e98c977b08bc%2F338d2349-d51c-4d57-a15e-9fcfbe6888a1%2Fdrgeig_processed.png&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(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 € X
such that f < 1.
3. Set En for all 0 ≤ n ≤ N and define S: X → RN+¹ by S(f) = (f(xo), f(x₁),..., ƒ(EN)).
=
(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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