Infinite Dimensional Banach Spaces

**raed** · March 23rd 2011, 05:50 AM

Dear Colleagues,

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

Regards,

Raed.

**girdav** · March 23rd 2011, 07:36 AM

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.

**raed** · March 23rd 2011, 11:46 AM

Thank you very much for your reply.

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