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.