0. Preliminary Results and Background

a. General notation. Let X, 7 b e Banach spaces . We will denote by B(X, Y)

the spac e o f al l bounde d operator s fro m X int o 7 , equippe d wit h it s natura l

norm. I n th e cas e X = 7 , w e simply writ e B(X) instea d o f B(X, X). W e will

denote by Id

x

th e identity operator on X, and by Bx th e unit ball of X. For all u

in B{ X, Y) an d all x i n X, we will often write ux instead of w(x) . The adjoint of

u will be denoted by u*. We will abbreviate finite-dimensional as f.d.

Let 1 / oo . B y a n L^-space , w e mea n a spac e /^(Af , 2, ra) fo r som e

measure spac e (M^,m). A s usual, we will denot e b y l

p

th e spac e R

n

(o r C

n

depending on the context) equipped with the norm

v(«,)€=R", ||(«,. ) || - ( E k i ' )1 * -

More generally, for any index set 7 , we denote by lp(I) th e space of all elements

x o f R

7

(o r C

7),

suc h tha t T,

ier\x(i)\p

oo, equipped wit h th e nor m

||JC| |

=

(L\x(i)\P)1^.

The spac e 1^(1) i s defined similarly , and c 0(I) i s defined a s the subspace of

1^(1) forme d by the elements x = (x(/))

/ e /

suc h that x(i) - 0 when / - oo . In

the case / = N , we write as usual l

p

or c0 instead of l p(I) an d c 0(I).

The class of all Z^-space s includes the Banach spaces lp, l p, l p(I) correspond -

ing to discrete measures on {1,2,...,«}, N or L We will also often work with the

class of C(#)-spaces—i.e., the class of Banach spaces of the form C(K) fo r some

compact set. Recall that any L^-space is isometric to a C(AT)-space, since it is a

commutative C*-algebra with unit.

The reader should recall that any Banach space is isometric to a subspace (resp.

a quotient) of an L^-spac e (resp. an L rspace).

By a subspac e o f a Banac h spac e X, w e mean (unles s specifie d otherwise ) a

closed subspace.

We say that a subspace E of X i s complemented i f ther e is a bounded linea r

projection P: X -* E (s o that X is isomorphic to E © KerP).

Let ( Xn) b e a sequenc e of Banac h space s and le t 1 p oo . We denote by

{Xx® X 2® - - • )

p

the space of all sequences (xn) i n n X

n

such that EH-XJI* oo.

We equip this space with its natural norm ||(x„)| | =

(L||JC

W||/)1//;

(wit h the usual

convention for p = oo) .

l

http://dx.doi.org/10.1090/cbms/060/01