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Weak continuity of the normalized duality map. (English) Zbl 1300.46011

Let \(X\) be a Banach space with dual \(X^*\). A gauge is a strictly increasing continuous functions \(\varphi:\mathbb R_+\to\mathbb R_+\) such that \(\varphi(0)=0\) and \(\lim_{t\to\infty}\varphi(t)=\infty.\) A duality mapping attached to the gauge \(\varphi\) is the set-valued mapping \(J_\varphi:X\rightrightarrows X^*\) given by \(J_\varphi(x)=\{x^*\in X^* : \langle x,x^*\rangle =\|x\|\,\|x^*\|,\, \varphi(\|x\|)=\|x^*\|\}\), see [I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems. Dordrecht etc.: Kluwer Academic Publishers (1990; Zbl 0712.47043)]. In the case \(\varphi(t)=t^{p-1}\) for some \(1<p<\infty\), the corresponding duality mapping is denoted by \(J_p\) and called the generalized duality map of order \(p\). In particular, \(J_2\) is denoted by \(J\) and is called the normalized duality mapping.
The paper is concerned with the weak continuity properties of the duality mapping \(J_p\) on \(\ell^p\) and on \(L^p=L^p[0,1]\), \(1<p<\infty\), meaning that \(x_n\to x\) weakly implies \(J_p(x_n)\to J_p(x)\) weak\(^*\). If \(X=H\) is a Hilbert space with the dual \(X^*\) identified with \(H\) through the Riesz representation theorem, then \(J(x)=x\), \(x\in H\), and so it is continuous in either the weak or the strong topology, but \(J_p\) is not weakly continuous for every \(p\in(1,\infty)\setminus\{2\}\). If \(p\in(1,\infty)\setminus\{2\}\), then the normalized duality mapping is not weakly continuous on \(\ell^p\), but the duality map \(J_p(x)=(|x_n|^{p-1}\mathrm{sign} (x_n))_{n\in\mathbb N}\), \(x=(x_n)\in\ell^p,\) is weakly continuous. The duality mapping on \(L^p, \) given by \(J_p(f)(t)=(\|f\|_p)^{-1}\,|f(t)|^{p-1}\mathrm{sign}(f(t))\) a.e. \(t\in[0,1]\), \(f\in L^p\setminus\{0\}\), is not weakly continuous.
The paper contains also some results on the duality mapping on product spaces as well as a short survey of its properties.

MSC:

46B10 Duality and reflexivity in normed linear and Banach spaces
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 0712.47043
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