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
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
th e spac e R
(o r C
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
(o r C
suc h tha t T,
oo, equipped wit h th e nor m
||JC| |
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
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 - - )
the space of all sequences (xn) i n n X
such that EH-XJI* oo.
We equip this space with its natural norm ||(x„)| | =
(wit h the usual
convention for p = oo) .
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