Bases of tensor products of Banach spaces

l Introduction* In this note we use the conventions and notations of Schatten [4] with the exception that we use B to indicate the dual (conjugate) space of a Banach space B and ζx, xy as the action of an element x and a functional x on each other. Schatten defines the tensor product Bi ® aB2 as the completion of the algebraic tensor product Bx ® B2 of two Banach spaces Bx and B2, on which the cross norm a has been imposed. We discuss the proposition, "If Bx and B2 have Schauder bases, then B1<^} aB2 has a Schauder basis/' We prove this for a = γ (Bλ 0 yB2 is the trace class of transformations of B[ into B2). We also prove it for a = λ {Bλ 0 λί?2 is the class of all completely continuous linear transformations of B[ into B2) in the case in which the bases of Bx and B2 satisfy an "isometry condition". This condition is not very restrictive. We know of no instance in which it is not satisfied. Next we show that unconditional bases of Bλ and B2 do not necessarily yield an unconditional basis for the tensor product, even in the nicest conceivable infinite dimensional case, that in which Bx = B2 = Hubert space, and the bases are orthonormal and identical. We recall certain facts about Schauder bases, and set some general notation that we use throughout the paper. We usually work with a biorthogonal set Ω — {xif #ί}{ associated with a Banach space B, so that X — {#;}; is a basis for B with coefficients supplied by the corresponding sequence of functionals χ' = {#!}». We will have to do with the closed linear manifold B of B generated by the elements of χ'. Since B and B are in duality it is possible to embed B in (B)' by the same formula that effects the embedding of B in B'\ We denote by n P m the projection of B defined by nPmx = Σ*T=n <#> <O&». The double sequence {nPm}%,m is uniformly bounded. We denote by T' the transpose of any transformation T. The following lemma, given without proof, is but a trivial strengthening of [2, p. 18, Theorem 1].