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Zbl 0779.46021
Franchetti, Carlo; Paya, Rafael
Banach spaces with strongly subdifferentiable norm. (English)
[J] Boll. Unione Mat. Ital., VII. Ser., B 7, No.1, 45-70 (1993). ISSN 0392-4041

Let $(X,\Vert \cdot \Vert)$ be a Banach space and let $D$ be the duality mapping from $S\sb X$, the unit sphere of $X$, to the subsets of the dual sphere $S\sb{X\sp*}$ of $X\sp*$. The strong subdifferentiability (ssd) of the norm in a point $u \in S\sb X$ was defined by {\it D. A. Gregory} [Can. Math. Bull., 23, 11-19 (1980; Zbl 0453.46032)] in the following formulation: the limit $$\lim\sb{t \to 0\sp +}{\Vert u+tx \Vert-1\over t}=\tau(u,x)$$ is uniform for $x\in B\sb X$, the unit ball of $X$. Also, the first author has introduced the notion of $\tau$-point [Arch. Math., 46, 76-84 (1986; Zbl 0564.46014)]: a point $u \in S\sb X$ is called $\tau$-point of $X$ if for any $K \ge 0$ there is a $t\sb K>0$ such that $$\inf \bigl\{1-\Vert u-tx\Vert;\tau(u,x) \le -1,\ \Vert x \Vert \le K \bigr\}>0.$$ We recall also that a point $u\in S\sb X$ strongly exposes the set $D(u)$ if $d(f\sb n,D(u))$ tends to zero for any sequence $(f\sb n)\subset B\sb{X\sp*}$ such that $f\sb n(u) \to 1$.\par The first main result of this paper establishes the equivalence of the following properties for an element $u \in S\sb X$.\par (i) $u$ strongly exposes the set $D(u)$;\par (ii) $D$ is norm-to-norm upper semicontinuous at $u$;\par (iii) for every $\varepsilon>0$ there is $\delta>0$ such that $d(D(x),D(y))<\varepsilon$ whenever $x\in S\sb X$ satisfies $\Vert x-u \Vert<\delta$;\par (iv) the norm of $X$ is ssd at $u$;\par (v) $u$ is a $\tau$-point of $X$.\par The second main result of this paper concerning stability properties of ssd of the norm under Banach space operations is the following: Let $\{X\sb \lambda;\lambda \in\Lambda\}$ be an arbitrary family of Banach spaces and let $C$ be the $\ell\sb \infty$-sum of such a family. For $u\in S\sb X$ let us write $\Lambda\sb u=\{\lambda \in \Lambda;\ \Vert u(\ )\Vert=1\}$. Then the norm of $X$ is ssd at $u$ if and only if the following two conditions are satisfied:\par (i) $\sup\bigl\{\Vert u(\lambda) \Vert;\ \lambda \notin \Lambda\sb u\bigr\}<1$ (in particular $\Lambda\sb u \ne \emptyset)$;\par (ii) for every $\varepsilon>0$ there is a $\delta>0$ such that $\lambda \in \Lambda\sb u$, $f\sb \lambda \in B\sb{X\sb \lambda\sp*}$ $\text{Ref} f\sb \lambda (u(\lambda))>1-\delta \Rightarrow d(f\sb \lambda,D(u(\lambda)))<\varepsilon$.\par Consequently, ssd of the norm is preserved under the formation of (arbitrarily-long) $c\sb 0$-sums.\par Also, the authors study dual properties of ssd and uniform ssd. Thus it is proved that the norm is ssd if and only if every $w\sp*$-exposed face in $B\sb{X\sp*}$ is strongly $w\sp*$-exposed and the norm of the dual is ssd if and only if the space is reflexive and every exposed face of $B\sb X$ is strongly exposed.\par Finally, the authors prove a statement formulated by G. Godefroy (private communication) that every Banach space with ssd norm is an Asplund space.
[T.Precupanu (Iaşi)]

MSC 2000:: *46B20 Geometry and structure of normed spaces
46B10 Duality and reflexivity in normed spaces
46B03 Isomorphic theory (including renorming) of Banach spaces

Keywords: $c\sb 0$-sums; $w\sp*$-exposed face; strong subdifferentiability of the norm; $\tau$-point; duality mapping; strongly exposes; stability properties; Asplund space

Citations: Zbl 0453.46032; Zbl 0564.46014

Cited in: Zbl 1197.46025 Zbl 0953.46009

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