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Let T : (0 → l defined by T(Xn)) = ( n. (a) Prove that TE B(l, l∞) and ||T|| = 1.

Let T : (0 → l defined by T(Xn)) = ( n. (a) Prove that TE B(l, l∞) and ||T|| = 1.

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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Let T : (0 → l∞ defined by T ((xn)) = (a) Prove that TE B(l,l∞) and ||T|| 1. (b) Show that Ran(T) is not closed in l. Hint. First show MÇ Ran(T) Ç Co for M {(y1,·… , Yn, 0, 0, · · · ) |n E N, y; E R}. Find Ker(T) and prove that dim(l) = dim(c) = dim(co).
Transcribed Image Text:Let T : (0 → l∞ defined by T ((xn)) = (a) Prove that TE B(l,l∞) and ||T|| 1. (b) Show that Ran(T) is not closed in l. Hint. First show MÇ Ran(T) Ç Co for M {(y1,·… , Yn, 0, 0, · · · ) |n E N, y; E R}. Find Ker(T) and prove that dim(l) = dim(c) = dim(co).
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