6 - Special Properties of c₀, l₁, and l_∞

Summary

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 l_p 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 l_p 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, (e_n) denotes the standard basis in c₀ or l₁, and denotes the associated sequence of coefficient functionals. As usual, (e_n) 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.