Strong subdifferentiability of convex functionals and proximinality
Let \(X\) be a real Banach space with dual \(X^*\) and let \(Y\) be a closed subspace of \(X\) with finite codimension \(n\). For \(x\in X\) and \(\delta> 0\) let \hskip 17mm \(d(x,Y)= \inf\{\|x-y\|; y\in Y\}\), \hskip 17mm \(P_Y(x)= \{y\in Y;\|x-y\|= d(x,Y)\}\), \hskip 17mm \(P_Y(x,\delta)= \{y\in Y;\|x-y\|< d(x, Y)+\delta\}\). The subspace \(Y\) is said to be \textit{strongly proximinal} if it is proximinal (i.e., \(P_Y(x)\) is nonempty for each \(x\in X\)) and the following holds: \[ \forall x\in X \forall\varepsilon> 0 \exists\delta> 0: y\in P_Y(x,\delta)\Leftrightarrow d(y, P_Y(x))< \varepsilon. \] A function \(F\) on some Banach space \(Z\) is said to be \textit{strongly subdifferentiable} at \(z\in Z\) if the one-sided limit \(\lim_{t\to 0^+}(F(z+ th)- F(z))/t\) exists uniformly for \(h\) in the unit sphere of \(Z\). It is well known that if \(Y\) is proximinal, then each functional in \(X^*\) that vanishes on \(Y\) attains its norm. The main result of the present paper (Theorem 3.2) says that \(Y\) is strongly proximinal if the support functions \(S_{C_i}: X^*\to \mathbb{R}\) of certain sets \(C_i\subset X\) \((i= 1,\dots, n)\) are strongly subdifferentiable at certain points. If, in particular, the norm of \(X^*\) is Fréchet differentiable at each nonzero norm-attaining functional of \(X^*\), then \(Y\) is strongly proximinal if and only if each functional in \(X^*\) that vanishes on \(Y\) attains its norm. The results are applied to the Banach space \(K(l_2)\) of compact operators on \(l_2\).
scientific article; zbMATH DE number 1800271
scientific article