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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
##### Keywords:
norm derivatives; Bishop-Phelps type theorem