Block-symmetric and weakly symmetric polynomials on Banach spaces of absolutely summable in a power $p\in [1,+\infty)$ sequences

Анотація

The work is devoted to the study of block-symmetric and weakly symmetric continuous polynomials on the Banach space $\ell_p^{ (\mathbb{K}) }$ of all sequences $x = (x_1, x_2, \ldots)$ with $x_j \in \mathbb{K}$ for $j\in\mathbb{N}$ such that the series $\sum_{j=1}^\infty |x_j|^p$ is convergent, where $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ and $p\in [1,+\infty).$
We construct an algebraic basis of the algebra of $n$-block-symmetric continuous polynomials on $\ell_p^{ (\mathbb{R}) }.$ Also we construct a generating system of the algebra of weakly symmetric continuous polynomials on $\ell_p^{ (\mathbb{K}) }$ for $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ and we show that elements of this system are linearly dependent.
We show that the weakly symmetric continuous linear functionals, in contrast to the block-symmetric functions, separate points of the space $\ell_1^{ (\mathbb{K}) },$ where $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}.$