Quojections and finite-dimensional decompositions

Abstract

K. Ftoret and V. B. Moscatelli proved in [5] that a Fr6chet space with an unconditional basis and without a continuous norm decomposes as a product of Fr6chet spaces with unconditional bases and continuous norms. This implies that a quojection with an unconditional basis is isomorphic to a product of Banach spaces with unconditional bases. In the same direction J. C. Diaz showed that no twisted quojection can have a complete lattice structure with the (weak)-Fatou property (cf. [2]). On the other hand, there are twisted quojections with a (conditional) basis, as shown in [8]. We present here an example of a twisted quojection with an unconditional finite-dimensional decomposition (FDD) in subspaces of dimension 2 (2-FDD in short) and with a basis. The method also applies to nuclear spaces and allows us to produce a nuclear Fr~chet space with a 2 -FDD which is not the product of spaces with continuous norms. This is done in Section 2. In Section 3 we show that a quojection with a strong F D D has a basis. For further information about quojections see the survey [9]. The notation here used is standard (cf. [6]) and we recall only some facts about finite-dimensional decompositions. Let F be a Fr6chet space and let (F,) be a sequence of closed subspaees of F. (F,) is a Schauder decomposition o f F if for every x in F there is a unique expansion x = 52 x,, with n x, e F,. If F, is finite-dimensional for any n, we say that (F,) is a finite-dimensional decomposition of F (FDD). We say that (F,) is a strong finite-dimensional decomposition of F (SFDD) if r = supdim(F,) < oe and, in this case, we say that (F.) is a r -FDD of F. If in the above definitions we require the unconditional convergence of the series x = Z x., we speak of unconditional Schauder decompositions, unconditional F D D and n unconditional SFDD.