On projections in power series spaces and the existence of bases

Abstract

Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace E E of a nuclear infinite type power series space, that its diametral dimension can be represented by Δ E = Δ Λ ∞ ( α ) \Delta E = \Delta {\Lambda _\infty }(\alpha ) for a suitable sequence α \alpha with α j ≥ ln ⁡ ( j + 1 ) {\alpha _j} \geq \ln (j + 1) . In this article we prove the existence of a basis for E E in case that α j ≥ j {\alpha _j} \geq j and sup α 2 j α j > ∞ \sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} > \infty .