Universality and the Szlenk Index

In this chapter, we study two closely related topics, namely the notion of the Szlenk index of a Banach space and the existence of universal Banach spaces with additional properties. Historically, this area of research arose from one of the problems from the Scottish book due to Banach and Mazur. The problem asked whether there exists a separable Banach (resp. reflexive separable) space that contains an isomorphic copy of every separable Banach (resp. reflexive separable) space. The first part was solved by Banach and Mazur themselves in the positive when they showed that C[0, 1] is a separable universal Banach space. The reflexive case was solved negatively by Szlenk using what is now called the Szlenk ordinal index of a Banach space. The value of the index of a separable reflexive space is a countable ordinal, and the index of a subspace is bounded from above by the index of the overspace. The negative solution of Szlenk then consisted of showing that there exists a separable reflexive space with an arbitrarily large countable index.

In the first two sections, we investigate various versions of the universality problem, with emphasis on reflexivity and complementability conditions. One of the main tools used is the theory of well-founded trees on Polish spaces. Using this notion, it is possible to assign to Banach spaces an ordinal index that measures the extent to which the space satisfies certain properties in which we are interested. One of them may be, for example, a containment of a certain subspace, but the technique is very versatile and includes the Szlenk index as a special case. The results include Bourgain’s theorem stating that a separable Banach space containing every separable reflexive space is universal for all separable spaces and the Prus, Odell, and Schlumprecht construction of a reflexive separable space universal for all separable superreflexive spaces.

The rest of the chapter is devoted to the development of the geometrical theory of the Szlenk index and its various applications to general universality problems, classification of C[0, α] spaces, and renormings. In particular, we prove the result of Bessaga, Pełczyński, and Samuel that the isomorphism type of C[0, α] for countable α is determined by the space’s Szlenk index. The connection to renormings is realized through the w*-dentability index Δ(X) and a theorem of Bossard and Lancien that claims the existence of a universal function \(\psi :\omega _1 \to \omega _1\) such that \(\Delta \left( X \right) \le \psi \left( {S_Z \left( X \right)} \right)\).


Banach Space Polish Space Separable Banach Space Separable Space Asplund Space 
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© Springer Science+Business Media, LLC 2008

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