Symmetric linear functionals on the Banach space generated by pseudometrics

In this work we consider the notion of B-equivalence of pseudometrics. Two pseudometrics d₁ and d₂ on a set X are called B-equivalent, where B is a subgroup of the group of all bijections on X, if there exists an element b of B such that d₁ (x, y) = d₂ (b (x), b (y) ) for every x, y X, that is, d₁ can be obtained from d₂ by permutating elements of X with the aid of the bijection b. The group B generates the group B of transformations of the set of all pseudometricson X, elements of which act as d (, ) d (b (), b () ), where d is a pseudometrics on X and b B. A function f on the set of all pseudometrics on Xis called B-symmetric if f is invariant under the action on its argument of elements of the group B. If two pseudometrics d₁ and d₂ are B-equivalent, then f (d₁) =f (d₂) for every B-symmetric function f. In general, the technique of symmetric functions is well-developed for the case of symmetric continuous polynomials and, in particular, for the case of symmetric continuous linear functionals on Banach spaces. To use this technique for the construction of B-symmetricfunctions on sets of pseudometrics, we map these sets to some appropriate Banach space V, which is isometrically isomorphic to the Banach space ₁of all absolutely summing real sequences. Weinvestigate symmetric (with respect to an arbitrary group of symmetry, elements of whichmap the standard Schauder basis of ₁ into itself) linear continuous functionalson ₁. We obtain the complete description of the structure of these functionals. Also we establish analogical results for symmetric linear continuous functionals on the space V. These results are used for the construction of B-symmetric functionals on the set of all pseudometrics on an arbitrary set X for the following case: the group B of bijections on X, that generates the group B, is such that the set of all x X, for which there exists b B such that b (x) x, is finite.