Let X be a normed space, z = X and f € X'.Show that T: X → X defined by T(x) = f(x)z,X e X is linear and compact.

the operator T : X -> X defined by T(x) = f(x) * z is linear, we need to verify two properties of…

Answered: Let X be a normed space, z = X and 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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Let V be the vector space of polynomials in & over R of degree <3 and define an inner product f. g = f₁¹ f(x)g(x)dxæ for all f, g € V. You may assume that this makes V into an inner product space. Define the linear map a: V→ V by a(f) (x) = f(x)+xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O a is self-adjoint O None of the others apply Ⓒa* = I-a, where I denotes the identity map O a is `anti-self adjoint' meaning a* = -a a is orthogonal
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Let X be a normed space, z = X and f € X'. Show that T: X → X defined by T(x) = f(x)z, xe X is linear and compact.