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Homework 12, Real Analysis

Due date: October 27


Problem 1


Let xnx,yny be convergent sequences in the normed space (X,||||) over \K. Then

  1. xn+ynx+y;

  2. λnxnμx for all sequences λnλ in \K.


Problem 2


Let (X,||||) be a normed space over \K.

  1. lim||xnx||=0 implies lim||xn||=||x||.

  2. For x,yX,λ\R, find lim(||(n+λ)x+y||||nx+y||).


Problem 3


Let X be a Banach space, xnX, ||xn||=1 for all n, and λn\K. Discuss the validity of the statement: λnxn converges if and only if |λn|<.

Problem 4


Let xn be a sequence in a Banach space X such that, for all \e>0, there exists a convergent sequence yn such that ||xnyn||<\e for all n.

  1. xn converges.

  2. Give an example where the previous statement is false if X is not complete.


Problem 5


The normed space (X,||||) is a Banach space if and only if the unit sphere

S={xX:||x||=1}


equipped with the metric d(x,y)=||xy|| is complete.

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