Bernstein's `lethargy' theorems in SF-spaces
Bernstein's `lethargy' theorem for Banach spaces states that given any strictly increasing sequence \(\{V_n\}\) of finite dimensional subspaces, for every sequence \(\varepsilon_n \downarrow 0\) there exists a \(x \in X\) such that \(\| x\| = \varepsilon_1\) and \(d(x,V_n) = \varepsilon_n\) for all \(n\). The following version of this theorem for SF spaces \((X,N)\) was obtained by \textit{G. Lewicki} [Monatsh. Math. 113, No.3, 213--226 (1992; Zbl 0764.41033)]: In addition to the hypothesis in the Banach space case, assume also \(R_N(\cup V_n) = \inf\{\sup~\{N(ty): t \geq 0\}: 0 \neq y \in \cup V_n \}\} >0\). Then for every sequence \(\epsilon_n \downarrow 0\) there exists a \(x \in X\) and \(n_0 \in \mathcal{N}\) such that \( d_N(x,V_n) = \epsilon_n\) for all \(n \geq n_0\). In this paper the author proves an analogue of the above result when \(V_n\)'s are generated by strictly increasing family \(\{K_n\}_{n \geq 1}\) of closed cones and obtains under some additional hypothesis a conclusion similar to the one above, now formulated in terms of \(K_n\).
scientific article; zbMATH DE number 1895110