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Continuous linear functionals and norm derivatives in real normed spaces. (English) Zbl 0787.46011
Let $$(X,\|\cdot\|)$$ be a real normed space, the norm derivatives are $(x,y)_{i(s)}= \lim_{t\to 0^{-(+)}}{\| y+tx\|^ 2- \| y\|^ 2\over 2t},\quad\text{for all } x,y\in X.$ $$f\in X^*$$ is called to be of (APP)-type if for any $$\varepsilon\in (0,1)$$ there exists a nonzero element $$y_ \varepsilon$$ in $$X$$ such that $|\langle y,y_ \varepsilon\rangle_ p|\leq \varepsilon\| y\|\cdot\| y_ \varepsilon\|,\quad\forall y\in\text{ker}(f),\quad p\in(i,s).$ In this paper, the author proves:
Theorem 1. Let $$X$$ be a Banach space, $$X$$ is reflexive iff for every continuous linear functional $$f$$ on $$X$$ there exists an element $$u$$ in $$X$$ such that $(x,u)_ i\leq f(x)\leq (x,u)_ s,\quad x\in X.$ Theorem 2. Let $$f$$ be a nonzero continuous linear functional of (APP)-type, then for any $$\varepsilon>0$$, there exists a nonzero element $$x_ \varepsilon$$ in $$X$$ such that $| f(x)- (x,x_ \varepsilon)|\leq \varepsilon\| x\|,\quad\forall x\in X,\quad p\in (i,s).$ {}.

##### MSC:
 46B20 Geometry and structure of normed linear spaces
##### Keywords:
(APP)-type functional; norm derivatives; reflexive