SYMMETRIC POLYNOMIALS ON CARTESIAN PRODUCTS OF REAL BANACH SPACES OF ABSOLUTELY SUMMABLE SEQUENCES - Varvariuk, M. M.; Vasylyshyn, T. V.; Nykorovych, I. I.; Ponomarov, R. V. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2025)
Let _p^(R) be the real Banach space of all absolutely summable in the power p[1,+∞) sequences of real numbers. For n, let _p_1^(R)×…× _p_n^(R) be the Cartesian product of Banach spaces _p_1^(R),…, _p_n^(R), where p_1,…,p_n[1,+∞). We consider the space _p_1^(R)×…× _p_n^(R) as the space of all sequences x= (x_1,x_2,…) of n-dimensional vectors x_j = ( x_j^(1),…,x_j^(n) ), where jN, such that the sequence ( x_1^(s), x_2^(s),… ) belongs to the space _p_s^(R) for every s{1,…,n}. A function f defined on the space _p_1^(R)×…× _p_n^(R) is called symmetric if f( ( x_σ(1),x_σ(2),… ) ) = f((x_1,x_2,…)) for every (x_1,x_2,…) _p_1^(R)×…× _p_n^(R) and for every bijection σ:N→N. We show that the set { H_p,k ^(): įnZ_+^n such that k_1/p_1+…+k_n/p_n ≥ 1 }