Chapter 3 - The James tree space JT

Summary

Creció en mi frente un árbol.

Creció hacia dentro.

Sus raíces son venas,

nervios sus ramas,

sus confusos follajes pensamientos.

Aliá adentro, en mi frente,

el árbol habla.

Acércate, ¿lo oyes? Octavio Paz

The simplest example of a separable Banach space with non-separable dual is l₁ and it was conjectured by Banach [1] that this was a sort of standard situation, namely that every separable Banach space with a non-separable dual had a subspace isomorphic to l₁. This raises the question of when a given non-reflexive Banach space does not admit l₁ or some other typical non-reflexive space as a subspace. In the previous chapter we saw that the space J has this kind of property, since neither c₀ nor l₁ can be embedded in J; however, J is not a counterexample to Banach's conjecture, because J* is separable. Nevertheless, using J as a building block, James [3] proved the conjecture to be false, by constructing the so-called James tree space JT, which is the subject matter of this chapter.

The space JT

Besides being the first example of a separable Banach space not containing l₁ with a non-separable dual (Lemma 3.a.5), the space JT has many other remarkable features, some shared with J and some not, and this makes it another important test case for many conjectures in the geometry of Banach spaces. Among the main properties of JT discussed in this section we cite the following: first, it is a somewhat reflexive space and even more, every infinite dimensional subspace contains an infinite dimensional Hilbert space, disproving another conjecture stated by Davis and Singer [1], who believed that each separable somewhat reflexive space had a separable dual.