Banach spaces containing copies of $\ell^1$

Question

Why is it important that a Banach spaces $X$ contains (or not) copies of the space $\ell^1$? I always hear talk about it but I don't know its importance.

Could someone explain this?

Answer 1

Containing $\ell^1$ implies that the space

1. Is non-reflexive (since a subspace of a reflexive space is also reflexive)
2. Has trivial type (=1), I won't go into details here.
3. Has nonseparable dual, since the dual of a subspace is a quotient of dual, and $(\ell^1)^*$ is nonseparable.

If you are interested in the geometry of Banach spaces, these are pretty interesting things to know about a space. If you are not, you can live your life without ever worrying about it.

Answer 2

I would say that it is important because of the Rosenthal's $\ell_1$-theorem which asserts that

Given a bounded sequence in a Banach space, it either contains a basic subsequence equivalent to a basis for $\ell_1$ or a weakly Cauchy subsequence.

Thus, if a space $X$ does not contain copies of $\ell_1$ all bounded sequences in $X$ have weakly Cauchy subsequences which is a nice and useful compactness-like statement. Note that we don't make any assumptions on $X$ whatsoever!