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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).

Advanced Engineering Mathematics
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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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.
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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