CHAPTER 0

INTRODUCTION

The purpose of this paper is to investigate the relationship among three ap-

parently different constructions of £p-spaces. We will show that two of the

methods produce the same isomorphic classes but that the third method pro-

duces a fundamentally different class of spaces. In particular the construction

due to Bourgain, Rosenthal and Schechtman [BRS] will be shown to produce dif-

ferent spaces than those Schechtman produced to show that there are infinitely

many isomorphically distinct £p-spaces. In order to explore the gap between

the two constructions we resurrect a 1974 construction of £p-spaces due to the

author [Al] that was presented in some seminars at Ohio State but was not pub-

lished at that time. (See [F] for a complete exposition and related results.) All

of the methods of construction make use of Rosenthal's fundamental space Xp

and thus have a probabilistic or distributional character that makes the passage

to the isomorphic level difficult. One consequence of this work is to show that

modifications of the ideas of Rosenthal can be used to work with these more

complex spaces within the isomorphic framework.

In Chapter 1 we will review the constructions and the basic properties. First

we will describe some of the results in Rosenthal's paper.

THEOREM

0.1. (Rosenthal's Inequality, [R,Theorem 3] or [JSZ],) Let 2 p

oo. Then there exists a constant Kp depending only on p such that if / i , . . . , fn

are independent, mean zero random variables in Lp, then

2max|(]C/\M dx) \J2j\Mdx)

/

(

pi n \P \

l/P

Ip'\

di

K

n

f\

\p

\

1 / / p

(

n

f\

I2

\

^T /

\fA dx)

, ( ] T /

\fA dx)

1/2

Using this inequality Rosenthal showed that there was a complemented sub-

space of Lp which he called Xp that was different (isomorphically) from the other

complemented subspaces known at the time. In its sequential form Xp^Wn) is

the completion of the space of sequences of real numbers (an) with only finitely

many an non-zero under the norm

, , oo \

I/P

/ oo v 1/2

N

| | K ) | | = maxj Q T K I ' J , (J2

\an\2™l)

}

1