Series expansions in fréchet spaces and their duals, construction of fréchet frames

Abstract

Frames for Fréchet spaces XF with respect to Fréchet sequence spaces ΘF are studied, and conditions implying series expansions in XF and XF* are determined. If {gi}i=1∞ is a Θ0-frame for X0 and ΘF (resp. XF) is given, we construct a sequence {Xs}s∈N{double-struck}0, Xs∈Xs-1, s∈N{double-struck}, (resp. {Θs}s∈N{double-struck}0, Θs∈Θs-1, s∈N{double-struck}), so that {gi}i=1∞ is a pre-F-frame or F-frame for XF with respect to ΘF under different assumptions given on X0, Θ0 and ΘF (resp. XF). © 2011 Elsevier Inc.