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 (Tzn)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 & > 0). 2. Show that the linear application T is continuous.
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 (Tzn)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 & > 0). 2. Show that the linear application T is continuous.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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