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H respectively. e. the matrix (S~ai, j» also represents a bounded linear operator D from X into Y. '=l are 1 then IIDII~ IITII. Proof. It is clearly enough to prove only the second part of the statement. Assume therefore that the unconditional constants are 1. Notice that the matrices A _ 1- ca. -a1,2 -a1,a a2,l a2,2 a2,a aa,l aa,2 aa,a ") ... A2 = c···· a1,2 a1,a -a2,l a2,2 a2,a -aa,l aa,2 aa,a ' ') ... represent operators with the same norm as that of T. , 21 c. Unconditional Bases represents an operator of norm ~ IITII.

P. (b) There is a separable Banach space having a Schauder basis whose dual is separable but fails to have the AP. Proof. 4. 1: {xn},~'=lcXand{x:}:'=lcX*aresuchthat co co 2: IIxnll IIx:11 <00 and n=1 2: x:(x)x,,=o, ,,=1 00 for every x E X, then also 2: ,,=1 x*(x,,)x:=O, for every x* E X*. In other words 00 2: n=1 Jx,,(x*)x: =0, where J: X ~ X** is the canonical embedding. Since X* has the AP. -l x:(xn ) =0. 35 e. P. d below). P. 3 there is a space Z so that Z** has a basis and Z**/Z is isomorphic to X.

3). For i=rn+j. O~r

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