Due date: October 27
Problem 1
Let xn→x,yn→y be convergent sequences in the normed space (X,||⋅||) over \K. Then
- xn+yn→x+y;
- λnxn→μx for all sequences λn→λ in \K.
Problem 2
Let (X,||⋅||) be a normed space over \K.
- lim||xn−x||=0 implies lim||xn||=||x||.
- For x,y∈X,λ∈\R, find lim(||(n+λ)x+y||−||nx+y||).
Problem 3
Let X be a Banach space, xn∈X, ||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 ||xn−yn||<\e for all n.
- xn converges.
- 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={x∈X:||x||=1}
equipped with the metric d(x,y)=||x−y|| is complete.
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