Algebraic bases of some algebras of polynomials on Banach spaces - Ponomarov, R. V.; Vasylyshyn, T. V. (Matematychni Studii, 2025)
The work is devoted to the study of algebraic bases of algebras of continuous polynomials on real and complex Banach spaces. A subset of an algebra is called an algebraic basis if every element of the algebra can be uniquely represented as a linear combination of products of powers of elements of the subset. Algebras of symmetric continuous polynomials on Banach spaces with some symmetric structure are typically equipped with finite or countable algebraic bases, which is important in the investigations of the respective algebras of symmetric analytic functions. Explicit constructions of such algebraic bases are often available when the Banach spaces are complex. In this work, we develop a method for extending these results to the case of real versions of such spaces. We also apply this method to the algebra of symmetric continuous polynomials on the Cartesian product of real Banach spaces of absolutely Lebesgue integrable in some powers functions on [0, 1].