This book deals with the following important aspect of modern analysis: Which Banach spaces contain almost isometric copies of one of the classical sequence spaces c0 or ^ for some 1 £/?<¥? (All Banach spaces here
are taken to be infinite dimensional.) A remarkable discovery of B. S. Tsirelson, in 1974, yields that there are
Banach spaces that contain no isomorphic copy of any of these spaces, while a recent discovery of E. Odell and
T. Schlumprecht asserts that Hubert space (i.e., ^) itself can be renormed to fail to have this property; that is,
the renormed space fails to contain an almost isometric copy of *». A deep result of D. Aldous, in 1980, proved
that every Banach space that isometrically embeds in Lp (for some 1 £/?<¥) has this property, and
subsequently J. L. Krivine and B. Maurey, in 1981, crystallized the important class of stable Banach spaces,
generalizing Aldous' proof, to show these also contain almost isometric copies of ^p for some 1 £ ρ < ¥. It is a
nontrivial result that Lp is stable, and moreover, Lp{X) is stable, for 1 £ ρ < ¥ and X a stable space. Thus every
subspace of Lp(Lq(Lr- · ·)) contains an almost isometric copy of some *p. It remains an open question, however,
if every subspace of a quotient of Lp con-