Let (en) be a complete orthonormal sequence in a Hilbert space H and let (2n) be a sequence of scalars. = (a) Show that there exists a unique operator T on H such that Ten λnen. (b) Show that T is bounded if and only if the sequence (2n) is bounded. (c) For a bounded sequence (2n), find the norm of T.
Let (en) be a complete orthonormal sequence in a Hilbert space H and let (2n) be a sequence of scalars. = (a) Show that there exists a unique operator T on H such that Ten λnen. (b) Show that T is bounded if and only if the sequence (2n) is bounded. (c) For a bounded sequence (2n), find the norm of T.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 46E
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![6. Let (en) be a complete orthonormal sequence in a Hilbert space H and
let (2n) be a sequence of scalars.
=
(a) Show that there exists a unique operator T on H such that Ten=
λnen.
(b) Show that T is bounded if and only if the sequence (λn) is bounded.
(c) For a bounded sequence (λn), find the norm of T.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2267b4ec-76c2-4b9e-b48b-65217f23fc7c%2Fb50d2cf6-c170-452d-8fd1-52a89134acec%2Firc809n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. Let (en) be a complete orthonormal sequence in a Hilbert space H and
let (2n) be a sequence of scalars.
=
(a) Show that there exists a unique operator T on H such that Ten=
λnen.
(b) Show that T is bounded if and only if the sequence (λn) is bounded.
(c) For a bounded sequence (λn), find the norm of T.
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