Best simultaneous approximation to totally bounded sequences in Banach spaces (Q944078)

Denote by \(\mathbb{R}^{m}(1\leq m\leq \infty )\) the Banach space of all real sequences \((m-\)tuples for\(\;m<\infty )\) with a monotonic norm \(\left\| \cdot \right\| .\) Let \((\lambda _{\nu })\in \mathbb{R}^{m}\;\)be fixed with each \(\lambda _{\nu }>0\) and let \((X,\) \(\left\| \cdot \right\| )\) be another Banach space over the field \(\mathbb{K}(=\mathbb{R}\vee \mathbb{C) }.\) One considers a fixed subset \(G\) of \(X\) and let \(\hat{x}=(x_{\nu })\) be a sequence in \(X\) such that \((\lambda _{\nu }\left\| x_{\nu }\right\| )\in (\mathbb{R}^{\infty },\left\| \cdot \right\| )\). The problem of best weighted simultaneous approximation to \(\hat{x}\) from \(G\) is the following: find \(g_{0}\in G\) such that \(\left\| (\lambda _{\nu }\right\| x_{\nu }-g_{0}\left\| )\right\| _{A}\leq \left\| (\lambda _{\nu }\right\| x_{\nu }-g\left\| )\right\| _{A},\) for all \(g\in G.\) The authors considers the problem of best weighted simultaneous approximation to totally bounded sequences in Banach space \(X.\) If \(X\) is uniformly smooth one obtain characterization results for the element of best weighted simultaneous approximation from a convex sets \(G\subset X,\) using functionals from the unit balls of \((\mathbb{R}^{\infty })^{\ast }\) and \(X^{\ast }.\)