Homework 12, Real Analysis
- Posted by Ricardo A. Sáenz
- on Oct, 20, 2017
- in Real Analysis, 2017
- Blog No Comments.
Due date: October 27
Problem 1
Let be convergent sequences in the normed space
over
. Then
;
for all sequences
in
.
Problem 2
Let be a normed space over
.
implies
.
- For
, find
.
Problem 3
Let X be a Banach space, ,
for all n, and
. Discuss the validity of the statement:
converges if and only if
.
Problem 4
Let be a sequence in a Banach space X such that, for all
, there exists a convergent sequence
such that
for all n.
converges.
- Give an example where the previous statement is false if X is not complete.
Problem 5
The normed space is a Banach space if and only if the unit sphere
equipped with the metric is complete.
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