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 $E$ of a nuclear infinite type power series space, that its diametral dimension can be represented by $\Delta E = \Delta {\Lambda _\infty }(\alpha )$ for a suitable sequence $\alpha$ with ${\alpha _j} \geq \ln (j + 1)$. In this article we prove the existence of a basis for $E$ in case that ${\alpha _j} \geq j$ and $\sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} < \infty$.