The Bishop-Phelps-Bollobás theorem on bounded closed convex sets (Q2801744)

This paper consists of four sections. In the first section, Introduction, the authors list some relevant results on the Bishop-Phelps-Bollobás property (BPBP), the definition of this property for linear operators between Banach spaces and also the BPBP on a bounded closed convex subset of \(X\) (a Banach space). In Section 2, it is shown that bounded linear functionals have the BPBP on arbitrary bounded closed convex sets. In particular, for a bounded closed convex set \(D\), the set \(\{f \in X^*: |f| \text{ attains its supremum on } D\}\) is dense in \(X^*\). The main result in the section reads as follows:NEWLINENEWLINETheorem. Let \(D\) be a bounded closed convex set in a Banach space \(X\). Given \(f \in X^*\) and \(\epsilon >0\), there exists \(x^* \in X^*\) such that \(|f+x^*|\) attains its supremum on \(D\) and \(\|x^*\|\leq \epsilon.\) Moreover, if \(D\) is symmetric and \(f(x_0)> \|f\|_D-\delta/2\) for some \(x_0 \in D\) and \(\delta >0\), then \(x^*\) and \(x_1 \in D\) can be chosen so that \(\|x^*\|\leq \epsilon\), \(\|x_0-x_1\|\leq \delta/\epsilon\) and \(|f+x^*|\) attains its suppremum at \(x_1\) on \(D\).NEWLINENEWLINEIn Section 3, the authors show several important results on pairs of spaces with the BPBP: {\parindent=0.6cm\begin{itemize}\item[--] \((X,Y)\) has the BPBP on every bounded closed convex subset \(D\) of \(X\), for all finite-dimensional Banach spaces \(X\) and \(Y\). \item[--] \((X,Y)\) has the BPBP on every bounded closed absolutely convex subset \(D\) of an arbitrary Banach space \(X\) and for a Banach space \(Y\) with Property \((\beta)\). \item[--] The pair \((X,Y)\) has the BPBP on a given bounded closed convex set \(D\) with positive modulus of convexity and for every Banach space \(Y\) . \item[--] \((X, C_0(L))\) has the BPBP on every bounded closed absolutely convex subset \(D\) of an Asplund space \(X\) and \(L\) is a locally compact Hausdorff topological space. NEWLINENEWLINE\end{itemize}} The last section deals with the stability of the BPBP on direct sums. It starts with a generalization of the \(\eta (\epsilon)\) function that appears in the definition of BPBP for a pair of spaces. More precisely, given two (real or complex) Banach spaces \(X\) and \(Y\), a bounded closed convex subset \(D\) of \(B_X\) and \(T \in \mathcal(X,Y)\), then NEWLINE\[NEWLINE \Pi_D(X,Y)=\{(x,T): x \in D, \|Tx\|=\|T\|_D=1\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \eta_D(X,Y)(\epsilon) = \inf\{ \|1-\|Tx\|: \, x \in D, \|T\|_D=1, \mathrm{dist} ((x,T), \Pi_D(X,Y))\geq \epsilon\},NEWLINE\]NEWLINE where \( \mathrm{dist} ((x,T), \Pi_D(X,Y)) =\inf \{\max \{ \|x-y\|, \|T-S\|\}: (y, S) \in \Pi_D(X,Y)\}\).NEWLINENEWLINEThe authors prove several propositions. The last proposition reads as follows:NEWLINENEWLINEProposition. Let \(X\) be a real Banach space containing a nontrivial \(L\)-summand, \(X=X_1\oplus_1 X_2\) for some nontrivial spaces \(X_1\) and \(X_2\), \(E_i\) denotes the natural isometric embedding of \(X_i\) in \(X\). Let \(D\) be a bounded closed convex subset of \(B_X\) such that \(D= \overline{co} (E_1D_1 \cup E_2 D_2)\) (where \(D_i\) is the image of \(D\) under the standard projection \(P_i\) of norm \(1\) from \(X\) onto \(X_i\)). If \(Y\) is a strictly convex space and if the pair \((X,Y)\) has the BPBP on \(D\), then \(Y\) is a uniformly convex space.