1. Le that if same lin 2. Co converg
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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1
![1. Let X be a Banach space an {n} be a sequence in. X. Prove
that if {Tn} converges in norm in X, then it converges weakly to the
same limit.
2. Consider the following sequences in l'. Determine which of them
converges weakly and find the weak limits when they exist.
a)
In = (1, ..., 1,0, .,).
b)
In = (1,, . .- 4, 0, ..).
2 3 * n'
c) In = (0, ..., 0, 1, 1,0, ...0, ...).
3. For which of the following space the unit ball is weakly compact
a) e;
b) e;
c) C[0, 1);
d) l;
e) L°[0, 1].
In each case explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e25041d-7573-46df-b9d3-ec2dd7694c16%2F568f8f8b-9977-432b-8c80-711500cb03d3%2Fdfrgbdf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let X be a Banach space an {n} be a sequence in. X. Prove
that if {Tn} converges in norm in X, then it converges weakly to the
same limit.
2. Consider the following sequences in l'. Determine which of them
converges weakly and find the weak limits when they exist.
a)
In = (1, ..., 1,0, .,).
b)
In = (1,, . .- 4, 0, ..).
2 3 * n'
c) In = (0, ..., 0, 1, 1,0, ...0, ...).
3. For which of the following space the unit ball is weakly compact
a) e;
b) e;
c) C[0, 1);
d) l;
e) L°[0, 1].
In each case explain your answer.
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