CHAPTER 1

PRESENTATIONS

We extract the relevant material from [3] and develop it further.

1.1. Admissible pairs of normed linear spaces and compatible pairs

of Banach spaces. Let X be a complex Banach space and X* a linear subspace

of the conjugate space, i.e., dual Xd of X. The X*-topology on X will be referred

to as the weak topology. Also the X-topology on X* will be referred to as the

weak*-topology.

1.1.1. We call (X, X*) an

ADMISSIBLE PAIR

of normed linear spaces if it

satisfies the assumptions, [3], §1:

(i) \\x\\ = sup{|p(x)| : p G X*, ||p|| 1} for x G X, and

(ii) the weakly closed convex hull of every weakly compact set in X is weakly

compact.

1.1.2. We identify a subset of a Banach space Z with its image in Zdd

under the canonical map r given by r(z)f = f(z) for z in Z and / in Zd. Given

an admissible pair (X, X*), assumption 1.1.1, (i) permits us to write X C Xd.

Thus (X*, X) is an admissible pair if and only if the following conditions are also

satisfied:

(hi) X* is closed in X d , and

(iv) the weak*-closed convex hull of every weak*-compact set in X* is weak*-

compact.

In this case we say that (X, X*) is a

COMPATIBLE PAIR

of Banach spaces.

1.1.3. Each one of (X, Xd) and (Xd, X) constitutes a compatible pair. This

follows from Krein Smulian theorem, cf., [22], p. 434.

1.2. Admissible pairs of spaces of operators. Let (X,X*) and (Y^Y*)

be admissible pairs. Let B(X, Y) and LW(X,Y) denote the Banach space of

bounded (linear) operators on X to Y and the linear space of all weakly continu-

ous operators on X to Y, respectively. For A G B(X, Y), x G X and p G Y*, A*p

and p®x are the bounded (linear) functionals on X and B(X,Y), respectively,

given by (A*p)(z) = p(Az) and p®x(T) = p{Tx). We have \\A*p\\ \\p\\ • \\A\\

and ||pEa;|| = ||p|| • ||rr||. For any operator A on X to Y, the symbols A*p and p8)x

will also be used for the functionals on X and LW(X, Y), respectively, defined by

the same rules. Let BW(X, Y) = B(X, Y) nLW(X, Y) and let BW(X, Y)* denote

the norm-closure in BW(X,

Y)d

of all finite sums Ylj Pj ® Xj-

In the space BW(X, Y) we use the adjective weak for the BW(X, Y)*-topology.

We remark that a norm-bounded net (Ap)p converges to A weakly in BW(X, Y)

if and only if for each x G X, the net (Apx)p converges to Ax weakly in Y if and

only if for each x G X, p £Y* the net ((p 0 x)(Ap))p = (p{Apx))p converges to

l