m at h . FA ] 2 6 O ct 1 99 3 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES

It is proved that there exist complemented subspaces of countable topo-logical products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces The problem of description of complemented subspaces of a given locally convex space is one of the general problems of structure theory of locally convex spaces. In investigations of this problem in the particular case of spaces represented as countable products of Banach spaces (see [D1], [D2], [DO], [MM1]) the following problem arose (see [D2, p. 71], [MM2, p. 147]): Is every complemented subspace of a topological product (locally convex direct sum) of a countable family of Banach spaces isomorphic to a topological product (locally convex direct sum) of Banach spaces? G.Metafune and V.B.Moscatelli [MM3, p. 251] conjectured that this is false, in general. The purpose of the present note is to prove this conjecture. Our sources for basic concepts and results of Banach space theory and the theory of topological vector spaces are, respectively, [LT] and [RR]. Let us fix some terminology and notation. The algebra of all continuous linear operators on a Banach space X will be denoted by L(X). The identity mapping of a linear space W is denoted by I W. Let {X n } ∞ n=1 be a sequence of Banach spaces. We denote their Cartesian product endowed with the product topology by ∞ n=1 X n and call it topological product. We denote the locally convex direct sum of spaces {X n } ∞ n=1 by ⊕ ∞ n=1 X n. A linear subspace Y of a topological vector space Z will be called complemented if there is a continuous linear mapping P of Z onto Y such that P 2 = P. If B is a subset of linear space V , then the linear subspace of V generated by B will be denoted by linB. The dual of a locally convex space Z endowed with its strong topology will be denoted by Z ′ β. Theorem 1. A. There exists a sequence {X n } ∞ n=1 of Banach spaces and a complemented subspace Y in X = ⊕ ∞ n=1 X n , such that Y is not isomorphic to a locally convex direct sum of Banach spaces. B. There exists a sequence {Z n } ∞ n=1 of Banach spaces and a complemented sub-space W in Z …