Nonlinear weighted best simultaneous approximation in Banach spaces (Q2473822)

Let \(\mathbb{R}^{\infty}\) be a Banach space of real sequences with a monotone norm \(\| .\| _A\) such that \[ \lim_{\nu \rightarrow \infty} \| (0,\dots,0,\lambda_{\nu},\lambda_{\nu+1},\dots) \| _A =0. \] Let \(\lambda_{\nu} = \frac{1}{(\nu^2 \| e^\nu \| _A)}\). Let \({\mathcal F} = \{\widehat{x}=(x_{\nu}): (\lambda_{\nu} \| x_{\nu}\| ) \in\mathbb R^{\infty}\}\), equipped with the norm \(\| \widehat{x}\| = \| (\lambda_{\nu} \| x_{\nu}\| )\| _A\). Let \({\mathcal F}_B = \{\widehat{x}=(x_{\nu}): (\| x_{\nu}\| )\) is bounded\}. Let \(X\) be a real or complex Banach space. A set \(G \subset X\) is said to be a BS-sun, if for any \(\widehat{x} \in {\mathcal F}_B\), \(g_0 \in P_G(\widehat{x})\Rightarrow g_0 \in P_G(\widehat{x}_{\alpha})\), where \(\widehat{x}_{\alpha} = g_0 + \alpha (\widehat{x}-g_0)\) for all \(\alpha>0\). One of the main results of this paper is to characterize BS-sun' as those for which \(g_0 \in P_G(\widehat{x})\) if and only if \(\max\{(a^\ast,(\Re\, \lambda_{\nu} f_{\nu}(g_0-g)))_A: (a^\ast,\widehat{f})\in E_{\widehat{x}-g_0}\} \geq 0\) for all \(g \in G\). Here \(E_{\widehat{x}-g_0}\) is an appropriate extremal set.