WEAKLY SYMMETRIC LINEAR CONTINUOUS FUNCTIONALS ON THE SPACE OF ABSOLUTELY SUMMABLE SEQUENCES

Abstract

The work is devoted to the study of weakly symmetric continuous linear functionals on the complex Banach space  of all absolutely summing complex sequences. In general, a function on a vector space is called symmetric with respect to some fixed group of operators on this space if the function is invariant under the action on its argument of elements of the group. A function on a vector space is called weakly symmetric with respect to some fixed descending by inclusion sequence of groups of operators on this space if this function is symmetric with respect to at least one of the groups that belong to the sequence. Spaces of symmetric continuous polynomials and, in particular, spaces of symmetric continuous linear functionals, on Banach spaces are complete with respect to the norm of the uniform convergence on the closed unit ball, which is one of the most commonly used norms on these spaces. In contrast, spaces of weakly symmetric continuous polynomials on Banach spaces with respect to the above-mentioned norm are not necessarily complete. Therefore, completions of these spaces can contain functions that do not satisfy any conditions of symmetry. Consequently, such functions can be approximated by weakly symmetric functions each of which, by the definition, is symmetric with respect to one of the above-mentioned groups. This fact makes it possible to apply to spaces of, in general, non-symmetric functions the technique developed for spaces of symmetric functions. In this work, we construct the sequence of groups of symmetries on the space . We obtain the structure of weakly symmetric, with respect to this sequence, continuous linear functionals on this space. Also, we find the completion of the space of all such functionals. Some properties of the completion are established.