PRELIMINARY RESULT S AN D BACKGROUN D
3
Let u s replac e X b y X* i n th e precedin g discussion . The n X* ® 7 ca n b e
identified wit h th e closur e i n B( X, Y) o f th e spac e o f al l finit e ran k operators .
(By bitransposition , eac h finit e ran k operato r u: X - Y defines a o(X**
9
X*)-
continuous operato r w**:^* * - * Y of finit e rank . Thi s correspondence i s clearly
isometric.)
To treat th e case of X* 0 Y similarly, we are lead to introduce the followin g
DEFINITION 0.1. A n operato r u: X - Y i s calle d nuclear i f i t admit s a
representation o f th e for m
00
(0.5) ux = £
x
;(x)y„ wit h x*
n
e ** , y„ e 7 ,
1
such tha t
Llk*||lklloo.
The nuclear norm of u is defined a s
AT(«) = inf|f ||x„*| | II^JlJ,
where the infimum run s over all representations of the form (0.5).
We wil l denot e b y N(X, Y) th e spac e o f al l nuclea r operator s u: X - Y
equipped with the norm N( •) .
Let J: X* ® Y - » Z * S Y b e th e natura l map . I t i s eas y t o sho w tha t
N(X,Y) i s linearly isomorphic to the image of / , J(X* ® 7) . MoreoverJV ( X, 7 )
is isometric t o th e quotien t X* ® 7/ker/ .
It i s important t o note tha t i f u: X - 7 i s a finite ran k operato r associate d t o
the tensor L" xf 0 y
i9
xf e Jf* , ^ e 7 , the n
1
w
^
1 1
X *
0
y\
In general , A^(w ) i s smalle r tha n ||E?x f ®^/||
A
. (Indeed , th e definitio n o f
HE"x* ® xf-||
A
is restricted t o representations of th e form (0.5 ) but fo r finite sum s
only!)
Grothendieck observe d tha t th e injectivity o f / : X 8 7 - Jf 0 7 fo r al l 7 , i s
equivalent t o th e so-calle d approximation property fo r X W e recall th e necessar y
definitions below .
DEFINITION
0.2. (i) Let X, 7 b e Banach spaces . An operato r u: X - 7 wil l be
called approximable i f it can be approximated uniforml y o n every compact subse t
of X b y finit e ran k operators . I n othe r words , fo r an y e 0 an d an y compac t
subset K o f X, ther e is a finite rank operator v: X - 7 suc h tha t
sup ||w x i;x|| e .
(ii) W e wil l sa y tha t u i s X-approximabl e i f ther e i s a ne t o f finit e ran k
operators v
f
: X - 7 , suc h tha t H^ H X , which converge s pointwise (or , equiva -
lently, uniformly o n the compact subset s of X) t o the operator u.
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