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Symmetric functions on spaces \(\ell_p(\mathbb{R}^n)\) and \(\ell_p(\mathbb{C}^n)\). (English) Zbl 1458.46033

Let \(n\in \mathbb{N}\), \(1\leq p<\infty \) and \(\mathbb{K=R}\) or \(\mathbb{K=C}\), let \(\ell _{p}(\mathbb{K}^{n})\) be the vector space of all sequences \(x=(x_{1},x_{2},\dots)\), where \(x_{j}=(x_{j}^{(1)},\dots,x_{j}^{(n)})\in \mathbb{K }^{n}\) for \(j\in \mathbb{N}\), such that \(\Vert x\Vert _{\ell _{p}(\mathbb{K}^{n})}=( \sum_{j=1}^{\infty }\sum_{s=1}^{n}\vert x_{j}^{(s)}\vert ^{p}) ^{\frac{1}{p}}<\infty \). The space \(\ell _{p}(\mathbb{K}^{n})\) with the norm \(\Vert \cdot\Vert _{\ell _{p}(\mathbb{K}^{n})}\) is a Banach space. The main objective of the paper is to construct an algebraic basis of the algebra of all continuous symmetric polynomials on \(\ell _{p}(\mathbb{R}^{n})\) and an algebraic basis of the algebra of all continuous symmetric \(\ast \)-polynomials on \(\ell _{p}(\mathbb{C}^{n})\).

MSC:

46G25 (Spaces of) multilinear mappings, polynomials
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46E25 Rings and algebras of continuous, differentiable or analytic functions
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References:

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