Dear Colleagues,
I have the following question,
Why infinite dimensional Banach spaces are not locally compact.
Regards,
Raed.
We only have to show that the unit ball of is not compact. If we assume that the unit ball is compact, then we can find an integer and such that .
If we denote by we have . Now we can show by induction that . is closed as a finite dimensional subspace hence and . is a finite dimensional space.