TY - JOUR AU - Kravtsiv, V.V. PY - 2019/06/30 Y2 - 2024/03/29 TI - Algebraic basis of the algebra of block-symmetric polynomials on $\ell_1 \oplus \ell_{\infty}$: Array JF - Carpathian Mathematical Publications JA - Carpathian Math. Publ. VL - 11 IS - 1 SE - Scientific articles DO - 10.15330/cmp.11.1.89-95 UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1511 SP - 89-95 AB - <p>We consider so called block-symmetric polynomials on sequence spaces $\ell_1\oplus \ell_{\infty}, \ell_1\oplus c, \ell_1\oplus c_0,$ that is, polynomials which are symmetric with respect to permutations of elements of the sequences. It is proved that every continuous block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ can be uniquely represented as an algebraic combination of some special block-symmetric polynomials, which form an algebraic basis. It is interesting to note that the algebra of block-symmetric polynomials is infinite-generated while $\ell_{\infty}$ admits no symmetric polynomials. Algebraic bases of the algebras of block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ and $\ell_1\oplus c_0$ are described.</p> ER -