Given an arbitrary infinite-dimensional separable complete linear metric space X, there exists a direct sum decomposition X = Va © V¡ such that each summand V¡ intersects every linearly independent Cantor set in X (this decomposition can be considered as a linear analogue to the classical Bernstein's decomposition into totally imperfect sets). Theorem. Each summand V of such a decomposition is not homeomorphic to its own square, and if T: V -> V is a linear bounded operator, then either the

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... el or the range of T is finite-dimensional. In the case of X -l2 this provides an example of a space V with the properties stated in the title, which answers a well-known question, cf. Arhangelskiï [A, Problem 21] and Geoghegan [G.Problem (LS 12)].