Consider the R-vector space 12 (R) := {("n) C R; <} n=0 of square summable reell sequences, endowed with the norm 1/2 Σ un u, n=0 Show that OB (R) (0, 1) = {(xn) E l2(R); ||(xn)||, (R) = 1} is closed and bounded, but not compact. Hint: Investigate the sequence (en), where en is the sequence whose n-th element is 1 and all others 0.
Consider the R-vector space 12 (R) := {("n) C R; <} n=0 of square summable reell sequences, endowed with the norm 1/2 Σ un u, n=0 Show that OB (R) (0, 1) = {(xn) E l2(R); ||(xn)||, (R) = 1} is closed and bounded, but not compact. Hint: Investigate the sequence (en), where en is the sequence whose n-th element is 1 and all others 0.
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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Transcribed Image Text:Task 14
Consider the R-vector space
12 (R) := {(xn) CR;
n=0
of square summable reell sequences, endowed with the norm
1/2
Show that dB, (R) (0, 1) = {(xn) E l2(R); ||(xn)|l12R) = 1} is closed and bounded, but not compact. Hint:
Investigate the sequence (en), where en is the sequence whose n-th element is 1 and all others 0.
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