Due date: October 27
Let be convergent sequences in the normed space over . Then
- for all sequences in .
Let be a normed space over .
- implies .
- For , find .
Let X be a Banach space, , for all n, and . Discuss the validity of the statement: converges if and only if .
Let be a sequence in a Banach space X such that, for all , there exists a convergent sequence such that for all n.
- Give an example where the previous statement is false if X is not complete.
The normed space is a Banach space if and only if the unit sphere
equipped with the metric is complete.