# Math Help - Infinite Dimensional Banach Spaces

1. ## Infinite Dimensional Banach Spaces

Dear Colleagues,

I have the following question,
Why infinite dimensional Banach spaces are not locally compact.

Regards,

Raed.

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