In this paper we determine theHausdorff measure of noncompactness on the sequence space $n\left( \phi \right)$ of $\text{W}\text{.}\,\text{L}\text{.}\,\text{C}.$ Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces $n\left( \phi \right)$ and $m\left( \phi \right)$. Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.

A Toeplitz decomposition of a locally convez space E into subspaces (Ek) with continuous projections (Pk) is a decomposition of every x ∈ E as x = ΣkPkx where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix. We extend to the setting of Toeplitz decompositions a number of results about the locally convex structure of a space with a Schauder decomposition. Namely, we give some necessary or sufficient conditions for being reflexive, a Montel space or a Schwartz space. Roughly speaking, each of these locally convex properties is linked to a property of the convergence of the decomposition. We apply these results to study some structural questions in projective tensor products and spaces with Cesàro bases.