The use of norm attainment (Q372701)

Let \(X\) be a Banach space and let \(S\) be a norm-closed subspace of its dual space \(X^*\). We denote by \(B_S\) its closed unit ball. If \(x^*(x) = 0\) for all \(x^* \in S\) implies that \(x = 0\), we call \(S\) separating. It is called 1-norming if, for every \(x \in X\), NEWLINE\[NEWLINE \| x \| =\sup \{| x^*(x) | : x^* \in B_S \}.NEWLINE\]NEWLINE The distinction between separating and 1-norming subspaces is related with the subset of \(X^*\) consisting of all functionals that attain their norm. Let us denote this set by NA\((X)\). To be specific, \textit{Yu. I. Petunin} and \textit{A. N. Plichko} [Ukr. Mat. Zh. 26, 102--106 (1974; Zbl 0297.46012)] showed that, if \(X\) is a separable Banach space and \(S\) a norm-closed separating subspace of \(X^*\) contained in NA\((X)\), then \(S\) is an isometric predual of \(X\) and in particular 1-norming.NEWLINENEWLINEThe author illustrates the versatility of this result and gives several examples belonging to various domains of analysis: special subspaces of \(L^1\), little Lipschitz spaces, weighted spaces of holomorphic functions, and spaces of compact operators.NEWLINENEWLINELet us give some details about the first one. Let \(X\) be a subspace of \(L^1\) whose unit ball is closed with respect to convergence in measure. Such subspaces are called ``nicely placed'' (see [\textit{G. Godefroy}, Trans. Am. Math. Soc. 286, 227--249 (1984; Zbl 0521.46012)]). Denote by \(X^\sharp\) the subspace of \(X^*\) consisting of the functionals whose restriction to \(B_X\) is continuous in the topology of convergence in measure. \(X^\sharp\) is contained in NA\((X)\), since every bounded sequence in \(L^1\) has a subsequence whose Cesàro averages converge in measure [\textit{J. Komlos}, Acta Math. Acad. Sci. Hung. 18, 217--229 (1967; Zbl 0228.60012)]. Thus \(X^\sharp\) is a predual of \(X\) as soon as it separates \(X\). This result was first shown by the author and \textit{D. Li} [Math. Scand. 66, No. 2, 249--263 (1990; Zbl 0687.46010)] by different methods, using the fact that there exists a closed subspace \(X_s\) of the bidual of \(L^1\) such that \((L^1)^{**} = L^1 \oplus_1 X_s\).