@article{Vasylyshyn_2020, title={Symmetric functions on spaces $\ell_p(\mathbb{R }^n)$ and $\ell_p(\mathbb{C }^n)$}, volume={12}, url={https://scijournals.pnu.edu.ua/index.php/cmp/article/view/3863}, DOI={10.15330/cmp.12.1.5-16}, abstractNote={<p>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 $1\leq p < +\infty.$</p> <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 $\ell_p(\mathbb{C}^n)$.</p> <p>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 $\ell_p(\mathbb{C}^n)$.</p>}, number={1}, journal={Carpathian Mathematical Publications}, author={VasylyshynT.V.}, year={2020}, month={Jun.}, pages={5-16} }