Abdeljawad, Thabet; Turkoglu, Duran; Abuloha, Muhib Some theorems and examples of cone Banach spaces. (English) Zbl 1198.46014 J. Comput. Anal. Appl. 12, No. 4, 739-753 (2010). Let \((E,P,\|\cdot\|)\) be an ordered Banach space where \(P\) is a closed, nonempty cone in \(E\). A cone normed space is an ordered pair \((X,\|\cdot\|_c)\), where \(X\) is a vector space over \(\mathbb R\) and \(\|\cdot\|_c:X\to (E,P,\|\cdot\|)\) is a function satisfying 6mm (1) \(0 \leq \|x\|_c , \forall x\in X\). (2) \(\|x\|_c = 0\Leftrightarrow x = 0\).(3) \(\|\alpha x\|_c = |\alpha|\,\|x\|_c\) for each \(x\in X\), \(\alpha\in\mathbb R\). (4) \(\| x+y \|_c\leq \| x\|_c + \| y \|_y\) \(\forall x,y \in X \). Each cone normed space is a cone metric space in the sense of L.-G. Huang and X. Zhang [J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)]. Given an ordered Banach space \((E,P,\|\cdot\|)\), a cone normed space structure is defined on \(E\) over itself. Thus a new norm \(\|\cdot\|_A\) is defined on \(E\). Various properties of this norm are studied. Reviewer: Şafak Alpay (Ankara) Cited in 19 Documents MSC: 46B40 Ordered normed spaces Keywords:ordered Banach space; cone normed space Citations:Zbl 1118.54022 PDFBibTeX XMLCite \textit{T. Abdeljawad} et al., J. Comput. Anal. Appl. 12, No. 4, 739--753 (2010; Zbl 1198.46014)