6

DUAL ALGEBRA S

that is weak*-convergent to 0 , then {$(x n)} is weak* convergent to 0 in Jt'.

Finally, if $ is weak* continuous and has range weak* dense in Jt, then there

exists a bounded, one-to-one, linear map £ : QM - X such that $ = j*, and if $

has trivial kernel and norm-closed range, then $(X*) = * Jt, $ is a weak*

homeomorphism of X* onto Jt, and *: Qjf - Xis invertible.

We shal l als o nee d frequentl y i n wha t follow s th e concep t o f th e absolutel y

convex hul l o f a se t £ i n a comple x vecto r spac e X. Th e se t E i s sai d t o b e

balanced i f XE c E fo r al l complex numbers A satisfying |X | 1. The absolutely

convex hull o f E, denote d b y aco(ls) , i s th e smalles t conve x an d balance d se t

containing E. Alternativel y i t i s th e collectio n o f al l linea r combination s a

l

xl

+ • • • +a nxnot vector s x v- • •, xn i n E suc h that \a x\ + • • * +|*J 1. If X i s a

Banach space , the n th e closed absolutely convex hull o f E, denote d b y aco(£) , i s

the norm-closur e o f aco(Z?) . Th e followin g proposition , whos e proo f ma y b e

found i n [17, Proposition 2.2], will be quite useful later .

PROPOSITION

1.21Let . X be a complex Banach space, let M be a positive

number, and let E be a subset of X. Then

W | M s u p | * ( x ) | , JeX* ,

if and only if aco(2s) contains the closed ball of radius \/M about the origin in X.

In particular, if E is a subset of the closed unit ball in X and

||*||« sup|*(x)| , 4GX*,

xeE

then aco(E) is the closed unit ball in X.