Compactness/convergence in Banach spaces

[TaPaKaH]

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

"every bounded sequence has a convergent subsequence"

holds in

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

Does it?

 

[micromass]

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.

 

[TaPaKaH]

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

 

[micromass]

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

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

 

[TaPaKaH]

Thank you!

 

[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.