Abstract. Given a subset A of a topological space X, a locally convex space Y , and a family C of subsets of Y we study the problem of the existence of a linear

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We prove that for each dense non-compact linear operator S : X → Y between Banach spaces there is a linear operator T : Y → c0 such that the operator T S : X → c0 is not compact. This generalizes the Josefson-Nissenzweig Theorem.By the Josefson-Nissenzweig Theorem [6], [7] (see also [2], [5, XII], and [3, 3.27]), for each infinitedimensional Banach space Y the weak * convergence and norm convergence in the dual Banach space Y * are distinct. This allows us to find a sequence (y * n ) n∈ω of norm-one functionals in Y * that converges to zero in the weak * topology. Such functionals determine a non-compact operator T : Y → c 0 that assigns to each y ∈ Y the vanishing sequence (y * n (y)) n∈ω ∈ c 0 . Thus each infinite-dimensional Banach space Y admits a non-compact operator T : Y → c 0 into the Banach space c 0 .The following theorem (which is a crucial ingredient in the topological classification [1] of closed convex sets in Fréchet spaces) says a bit more:Theorem 1. For any dense non-compact operator S : X → Y between Banach spaces there is an operator T : Y → c 0 such that the composition T S : X → c 0 is non-compact.1991 Mathematics Subject Classification. 47B07; 46B15.