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Exercise 4 Let (fn) be a sequence of functions defined by In(2)=√(n.2n}(2). 1. Prove that (fa) is weakly convergent to 0 in L2([0, +∞0)), but does not converge strongly in L²([0, +∞)). 2. Prove that (fn) converge strongly in LP ([0, +00)), for p > 2.

Exercise 4 Let (fn) be a sequence of functions defined by In(2)=√(n.2n}(2). 1. Prove that (fa) is weakly convergent to 0 in L2([0, +∞0)), but does not converge strongly in L²([0, +∞)). 2. Prove that (fn) converge strongly in LP ([0, +00)), for p > 2.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 34E
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Exercise 4 Let (fn) be a sequence of functions defined by fn (2)=√(n.2n) (20). 1. Prove that (fm) is weakly convergent to 0 in L2([0, +∞0)), but does not converge strongly in L²([0, +∞)). 2. Prove that (fn) converge strongly in LP([0, +00)), for p > 2.
Transcribed Image Text:Exercise 4 Let (fn) be a sequence of functions defined by fn (2)=√(n.2n) (20). 1. Prove that (fm) is weakly convergent to 0 in L2([0, +∞0)), but does not converge strongly in L²([0, +∞)). 2. Prove that (fn) converge strongly in LP([0, +00)), for p > 2.
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