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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
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