The Equation L(E, X ∗ ∗ = L(E, X) ∗ ∗ and the Principle of Local Reflexivity

Abstract

A new derivation of the equation L(E, X**)= L(E, X)* * is given, for dim(E) T in the weak-star operator topology). LEMMA 1. Let (Ta) be a net in L(X, Y) and T in L(X, Y**) with 11 Tall _ 11 T)) for each o. Suppose A C X, TA C Y and w*-op lima T,= T. Then, to 8>0, there is a net (Sa)cL(X, Y) such that JIS,II<IITII+8, w*-op lim S, =Tand Saa=Ta for each a in A. PROOF. For each a in A, (Taa) converges weakly to Ta. Since dim(A)< oo, using standard techniques (e.g. [2, p. 477]), a net of convex combinations of (Ta), say (Ua), converges in norm on A, and w*-lim Ua,7=T. Write X=A Z and set Sj(a+f)= Ta+ Uj. Then Sa e L(X, Y) and Sa-UI O-+0. Thus, for large o, 1Sa1 < 1 TII + E. If dim(E)= 1, then L(E, X)**=L(E, X**) is simply the statement that the unit ball U1(X)={x, IIxlIl1} is weak-star dense in U1(X**). As Received by the editors December 1, 1972. AMS (MOS) subject class/ifcations (1970). Primary 46B10, 47A65.