space Suppose that (Tx, x) > 0 for all x H. 1. Let (zn)n1 be a sequence of H that converges to 0. Suppose that (Tn)n>1 converge to LEH. (a) Show that (L, h) + (Th, h) >0 for all h € H. (b) Deduce that L=0. (Hint: replace h by sh for all e > 0).
space Suppose that (Tx, x) > 0 for all x H. 1. Let (zn)n1 be a sequence of H that converges to 0. Suppose that (Tn)n>1 converge to LEH. (a) Show that (L, h) + (Th, h) >0 for all h € H. (b) Deduce that L=0. (Hint: replace h by sh for all e > 0).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
Related questions
Question
hilbert space part 1 operator
![Problem 1.
Let H be a Hilbert space and T: H → H be a linear application.
Suppose that (Tx, x) > 0 for all x € H.
1. Let (zn)n1 be a sequence of H that converges to 0. Suppose that (Tn)n>1 converge
to LE H.
(a) Show that (L, h) + (Th, h) >0 for all h & H.
(b) Deduce that L = 0. (Hint: replace h by ch for all e > 0).
2. Show that the linear application T is continuous.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13032172-d002-48c5-a8a1-534065d0933a%2F2d69bf43-e94d-4c2f-a054-3e05a21e6db4%2Ftdz6t8k_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1.
Let H be a Hilbert space and T: H → H be a linear application.
Suppose that (Tx, x) > 0 for all x € H.
1. Let (zn)n1 be a sequence of H that converges to 0. Suppose that (Tn)n>1 converge
to LE H.
(a) Show that (L, h) + (Th, h) >0 for all h & H.
(b) Deduce that L = 0. (Hint: replace h by ch for all e > 0).
2. Show that the linear application T is continuous.
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