On the associate and conjugate space for the direct product of Banach spaces

The direct product Ei®nE2 of two Banach spaces Eu E2 has been defined before [5](2) as the closure of the normed linear set $In(Ei, E2) (that is, linear set 3i(£i, £2) of expressions 22Â-ifi®4>i, hi which N is a norm) [5, p. 200, Definition 1.3] and [6, p. 499, b]. Let N denote a crossnorm whose associate N' is also a crossnorm [5, p. 208]. Then, the cross-space Ei®nEi determines uniquely a "conjugate space" (Ei®NE2)' and an "associate space" Ei ®n'E2 . It is shown [5, p. 205] that Ei ®N'E2 is always included in (Ei®nE2)'. While there are many known examples of cross-spaces for which the associate space coincides with the conjugate space—for example, the cross-space generated by the self-associate crossnorm constructed for Hubert spaces by F. J. Murray and John von Neumann [3, p. 128] and [5, pp. 212-214]—it is not without interest to construct a cross-space for which the associate space forms a proper subset of the conjugate space (§§1-2). For reflexive Banach spaces Ei, E2 (that is, such that 25/' =Ei), and a reflexive crossnorm N [6, p. 500], the reflexivity of Ei®^E2 implies (Ei®nE2)'—Ei ®N'Ei [6, p. 505]. Thus, the finding of the exact conditions imposed upon reflexive Banach spaces and a reflexive crossnorm for which the resulting cross-space is reflexive is closely connected with the above-mentioned problem. In §1, we show that for a "natural crossnorm" N, L'®nL' is a proper subset of (L®NL)'. In §2 we prove that for a "natural crossnorm" N, V®ud' is a proper subset of (/ ® nI) '• In §3 we show that for any p > 1, lp ® nIi is not reflexive, provided l/p + l/q = 1 and N denotes the least crossnorm whose associate is also a crossnorm [5, p. 208]. The last one is reflexive [6, p. 501 ]. 1. Let Z,(i) and L$) denote the Banach spaces of all functions integrable in the sense of Lebesgue on the interval Ogsgl, and on the square 0^s, t ^ 1 respectively. Similarly, let M(u and M denote the Banach spaces of all functions Lebesgue measurable and essentially bounded on the interval OiSs^l and the square 0^s, f gl respectively [l, pp. 10, 12]. We recall that