Let T: C[0, 1] → C[0, 1] be defined by= [² fTf(x) =f(t)dt.(a) Show that T is linear.(b) Find T-¹: R(T) → C[0, 1], the inverse of T.(c) Determine if T-¹ is linear and bounded? (Hint: Consider function of polynomialsdegree n.)(d) Determine Ker(T-¹).

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Answered: Let T: C[0, 1] → C[0, 1] be defined by…

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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Let T: C[0, 1] → C[0, 1] be defined by Tf(x) = f f(t)dt. (a) Show that T is linear. (b) Find T-¹: R(T) → C[0, 1], the inverse of T. (c) Determine if T-¹ is linear and bounded? (Hint: Consider function of polynomials degree n.) (d) Determine Ker(T-¹).