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: XC([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 ƒ € X such that f. 1. 3. Set n = 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.

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: XC([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 ƒ € X such that f. 1. 3. Set n = 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.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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