Some Realisation of the Banach Space of All Continuous Linear Functionals on ℓ1 Approximated by Weakly Symmetric Continuous Linear Functionals

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is ℓ1 (i.e., the space of all absolutely summable sequences of complex numbers), reduces to a continuous linear functional whose Riesz representation is a periodic sequence. We consider the completion of the space of all such linear continuous functionals on ℓ1 with periods of Riesz representations equal to powers of 2. It is known that this completion is a Banach space with a Schauder basis. In this work, we construct a sequence Banach space with the standard Schauder basis {em=(0,…,0︸m−1,1,0,…)}m=1∞ that is isometrically isomorphic to this completion. Results of the work can be used to describe spectra of topological algebras of analytic functions on ℓ1 that can be approximated by weakly symmetric functions.