Approximation of continuous linear functionals in real normed spaces. (English) Zbl 0787.46012

Dragomir, S. S.

Rend. Mat. Appl., VII. Ser. 12, No. 2, 357-364 (1992).

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 \[ -\varepsilon\| x\|+ (x,u_{f,\varepsilon})_ i\leq f(x)\leq (x,u_{f,\varepsilon})_ s+ \varepsilon\| x\|. \] {}.

Reviewer: Yu Xintai (Shanghai)

MSC:

46B20

Geometry and structure of normed linear spaces

Keywords:

norm derivatives; Bishop-Phelps type theorem