Weak-polynomial convergence on spaces \(\ell_{p}\) and \(L_{p}\)
The authors consider various topologies on the real Banach spaces \(X=\ell_p\) or \(X=L_p[0,1]\), \(1 \leq p < \infty\). The following two topologies are in the center of investigations. The Banach space of all scalar-valued continuous \(n\)-homogeneous polynomials on \(X\) is denoted by \(P(^nX)\) and \(P_o(^nX)\) denotes the closed subspace of orthogonally additive polynomials, where the orthogonality results from the lattice structure of \(X\). For \(P \in P(^nX)\), define \(\widetilde{d}_P:X \times X \rightarrow {\mathbb R}\) by \[ \widetilde{d}_P(x,y) = \inf \{ | P(x-z_1)| ^{1/n} + | P(z_1-z_2)| ^{1/n} + \cdots + | P(z_{k-1}-z_k)| ^{1/n} + | P(z_k-y)| ^{1/n} \}, \] where the infimum is taken over all \(z_1,\dots, z_k \in X, k \geq 1 \), and put \(d_P(x)=\widetilde{d}_P(x,0)\). Then the topology \( \tau \) on \(X\) is defined by the family of seminorms \((d_P)_P\) when \(P\) varies over \(P_o(^nX)\) and \(n \geq 1\). A net \((x_\alpha)\) in \(X\) converges to \(x \in X\) with respect to the topology \(wp_0\) if \(P(x-x_\alpha) \rightarrow 0\) for all \(P \in P_o(^nX)\). The main results are the following. Theorem: Let \(X=\ell_p\), \(1\leq p < \infty\), and let \(k\) be the smallest integer with \(p \leq 2k\). Then for any bounded net \((x_\alpha)\) and \(x\) in \(X\) it is satisfied that \( x_\alpha \rightarrow x \) w.r.t. \(\tau\) iff \( x_\alpha \rightarrow x \) weakly and \(\| x_\alpha - x \| _{2k}\rightarrow 0\). Corollary: Let \(X=\ell_p\), \(p=1\) or \(p \in (2k-1,2k]\) for some integer \(k\). Then for any bounded net \((x_\alpha)\) and \(x\) in \(X\) it is satisfied that \( x_\alpha \rightarrow x \) w.r.t. \(\tau\) iff \( x_\alpha \rightarrow x \) w.r.t. \(wp_o\). Theorem: Let \(X=L_p[0,1]\), \(1\leq p < \infty\) and let \(k\) be the largest integer with \(2k \leq p\). Then for any bounded net \((x_\alpha)\) and \(x\) in \(X\) it is satisfied that \( x_\alpha \rightarrow x \) w.r.t. \(\tau\) iff \( x_\alpha \rightarrow x \) weakly and \(\| x_\alpha - x \| _{2k}\rightarrow 0\). Corollary: Let \(X=L_p[0,1]\), \(1 \leq p < 2\) or \(p \in [2k,2k+1)\) for some integer \(k\). Then for any bounded net \((x_\alpha)\) and \(x\) in \(X\) it is satisfied that \( x_\alpha \rightarrow x \) w.r.t. \(\tau\) iff \( x_\alpha \rightarrow x \) w.r.t. \(wp_o\). In order to prove these results, the authors show various results concerning \(d_P\), using \(P^{-1}(0)\) as the crucial point.
zeros of polynomials on \(\ell_p\) and \(L_p\)
scientific article; zbMATH DE number 2145886
scientific article