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Approximation of continuous linear functionals in real normed spaces. (English) Zbl 0787.46012
Let \(X\) be a Banach space, the norm derivatives are \[ (x,y)_{i(s)}= \lim_{t\to 0^{-(+)}} {\| y+ tx\|^ 2-\| y\|^ 2\over 2t},\quad\forall x,y\in X. \] In this paper, the author proves the Bishop- Phelps type theorem:
Let \(X\) be a real Banach space. Then for every continuous linear functional \(f: X\to R\) and for any \(\varepsilon>0\) there exists an element \(u_{f,\varepsilon}\) such that \[ -\varepsilon\| x\|+ (x,u_{f,\varepsilon})_ i\leq f(x)\leq (x,u_{f,\varepsilon})_ s+ \varepsilon\| x\|. \] {}.

MSC:
46B20 Geometry and structure of normed linear spaces
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