On the existence of a basis in a complemented subspace of a nuclear Köthe space from class $(d_1)$

Exposé on a paper of Dronov and Kaplitzkii

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In [1] Dronov and Kaplitzki showed that every complemented subspace of a nuclear Köthe space E with a regular basis of type (d 1 ) has a basis so, in particular, solving the long standing problem whether any complemented subspace of the space (s) of rapidly decreasing sequences has a basis. We present a slightly modified version of their proof which shows that the range of every closed-range operator in E has a basis.

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“…By use of the interpolation theorem for cones [1,Theorem 1] we obtain that for every r there is a constant C(r) such that…”