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Banach Spaces

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Show that (X, ||*||) is a Banach space if and only if {x in X: ||x||=1} is complete.

Know that in the first direction, we must show that {x in X: ||x||=1} is closed subset of X.

For the reverse direction, I know I have to take a cauchy sequence and translate it to the unit circle and then show that if it is convergent there, it is convergent outside of the unit circle.

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Solution Summary

Banach spaces are investigated thoroughly in the solution.

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Proof:
"=>": Suppose (X,||.||) is a Banach space. We are trying to show that
the set C={x in X: ||x||=1} is complete. Actually, we only need
to show that C is closed. We consider a sequence {x_n} in C and
x_n->x. I claim that x is also in C.
Since X is a Banach space, then x is in X. Since x_n->x, then for
any e>0, we can find some N>0, such that for all n>N, we have
||x_n-x||<e. We note the triangle inequality
| |a|-|b| | <= |a-b|
Then we have |||x_n||-||x|||<=||x_n-x||<e, we also note ||x_n||=1,
then ...

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