On some generalized triangle inequalities and \(\ell_{\psi}\)-spaces. (English) Zbl 1352.46023

Summary: In this paper, we consider a generalized triangle inequality of the following type: \[ \| a_1 x_1+\cdots + a_1 x_n \| ^p\leq \| x_1\|^p +\cdots +\| x_n\| ^p \;(x_1,\dots, x_n\in X), \] where \((X, \| \cdot \|)\) is a normed space, \((a_1, \dots, a_n) \in \mathbb C^n\) and \(p>0\). By using generalized \(\ell_p\)-spaces, we present a characterization of above inequality for infinite sequences \(\{x_n\}_{n=1}^{\infty} \subset X\).


46B99 Normed linear spaces and Banach spaces; Banach lattices
Full Text: Euclid


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