The spaces c0, L1, L2, and L∞; play very special roles in Banach space theory. You're already familiar with the space l and its unique position as the sole Hilbert space in the family of lp spaces. We won't have much to say about l2 here. And by now you will have noticed that the space l∞ doesn't quite fit the pattern that we've established for the other lp spaces. for one, it's not separable and so doesn't have a basis. Nevertheless, we will be able to say a few meaningful things about l∞. The spaces c0 and l1, on the other hand, play starring roles when it comes to questions involving bases in Banach spaces and in the whole isomorphic theory of Banach spaces for that matter. Unfortunately, we can't hope to even scratch the surface here. But at least a few interesting results are within our reach.

Throughout, (en) denotes the standard basis in c0 or l1, and denotes the associated sequence of coefficient functionals. As usual, (en) and are really the same; we just consider them as elements of different spaces.

True Stories Aboutl1

We begin with a “universal” property of l1 due to Banach and Mazur [8].

Theorem 6.1.Every separable Banach space is a quotient of l1.

Proof. Let X be a separable Banach space, and write = {x : ∥ x ∥ < 1} to denote the open unit ball in X.