On some generalized triangle inequalities and $\ell_{\psi}$-spaces

Abstract

In this paper, we consider a generalized triangle inequality of the following type: \begin{equation*} \Vert a_1 x_1+\cdots + a_1 x_n \Vert ^p \leq \Vert x_1\Vert^p +\cdots +\Vert x_n\Vert ^p \ (x_1,\ldots, x_n \in X ), \end{equation*} where $(X, \Vert \cdot \Vert)$ is a normed space, $(a_1, \ldots, a_n) \in \Bbb C^n$ and $p>0$. By using generalized $\ell_p$-spaces, we present a characterization of above inequality for infinite sequences $\{x_n\}_{n=1}^{\infty} \subset X$.