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 $\displaystyle \overline B$ of $\displaystyle E$ is not compact. If we assume that the unit ball is compact, then we can find an integer $\displaystyle N$ and $\displaystyle x_1,\ldots,x_N \in\overline B$ such that $\displaystyle \displaystyle \overline B\subset\bigcup_{j=1}^NB(x_j,\frac 12)$.
If we denote by $\displaystyle F=\mathrm{Span}(x_1,\ldots,x_N)$ we have $\displaystyle \overline B \subset F+B\left(0,\frac 12\right)$. Now we can show by induction that $\displaystyle \overline B\subset F+B\left(0,\frac 1{2^n}\right)$. $\displaystyle F$ is closed as a finite dimensional subspace hence $\displaystyle \overline B\subset F$ and $\displaystyle E=F$. $\displaystyle E$ is a finite dimensional space.