# Compactness/convergence in Banach spaces

1. Oct 12, 2012

### TaPaKaH

Been doing exercises on compactness/sequential compactness of objects in Banach spaces and some of my solutions come down to whether
holds in

a) arbitrary finite-dimensional Banach space
b) lp, 1 <= p <= infinity

Does it?

2. Oct 12, 2012

### micromass

Staff Emeritus
No, this does not hold. Consider the sequence in $\ell^2$:

$$x_1=(1,0,0,0,...)$$
$$x_2=(0,1,0,0,...)$$
$$x_3=(0,0,1,0,...)$$
and so on.

3. Oct 12, 2012

### TaPaKaH

Doesn't hold for finite-dimensional Banach space as well?

4. Oct 12, 2012

### micromass

Staff Emeritus
It does hold for finite dimensional spaces since those are isomorphic to $\mathbb{R}^n$.

Last edited: Oct 12, 2012
5. Oct 12, 2012

### TaPaKaH

Thank you!

6. Oct 13, 2012

### Bacle2

In general, in metrizable spaces, compactness and sequential compactness are

equivalent. The unit ball is compact/seq. compact in a normed v.space V iff

V is finite-dimensional.