Ex3LetT:C ([0,1]) → Rf→ f(0)i)If C ([0,1]) is equipped with the sup-norm, show that T is boundedand compute its norm.ii)If C ([0,1]) is equipped with the L¡ -norm , show that T is unbounded.Ex4:Let X be a normed linear space.Use the Hahn-Banach theorem to prove the following statements.i)For any non zero x in X, there is a bounded linear functionalO in X such thatIl|| = 1 and ø(x) = ||x||.ii) If x, y in X verify (x) = ¢(y) ▼ ¢ in X' , then x = y.

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Answered: Ex3 i) ii) Let T: C ([0,1]) → R f →…

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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Ex3 i) ii) Let T: C ([0,1]) → R f → f(0) If C ([0,1]) is equipped with the sup-norm, show that T is bounded and compute its norm. If C ([0,1]) is equipped with the L₁ -norm, show that T is unbounded.