Symmetric functions on spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ - Vasylyshyn, T.V. (Carpathian Mathematical Publications, 2020)
This work is devoted to the study of algebras of continuous symmetric polynomials, that is, invariant with respect to permutations of coordinates of its argument, and of $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers respectively, where $1≤ p < +∞.$ We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$. Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n)$. Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$, and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n)$.