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Another approach to characterizations of generalized triangle inequalities in normed spaces. (English) Zbl 1311.46014

Summary: In this paper, we consider a generalized triangle inequality of the following type: \[ \| x_1+\cdots+x_n\|^p\leq\frac{\| x_1\|^p}{\mu_1}+\cdots+\frac{\| x_2\|^p}{\mu_n} \quad \text{(for all }x_1,\dots,x_n\in X), \] where \((X,\|\cdot\|)\) is a normed space, \((\mu_1,\dots,\mu_n)\in\mathbb R^n\) and \(p>0\). By using \(\psi\)-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which has been given by F. Dadipour et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 2, 735–741 (2012; Zbl 1242.46029)].

MSC:

46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory

Citations:

Zbl 1242.46029
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References:

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