Supporting vectors for the ℓ1-norm and the ℓ∞-norm and an application

Abstract

A supporting vector of a matrix A for a certain norm $\Vert ·\Vert$ on ${\mathbb{R}}^n$ is a vector x such that $\Vert x\Vert =1$ and ${\displaystyle \Vert Ax\Vert =\Vert A\Vert =\underset{\Vert y\Vert =1}{max}\Vert Ay\Vert }$. In this manuscript, we characterize the existence of supporting vectors in the infinite-dimensional case for both the ${\ell }_1$-norm and the ${\ell }_{\infty }$-norm. Besides this characterization, our theorems provide a description of the set of supporting vectors for operators on ${\ell }_{\infty }$ and ${\ell }_1$. As an application of our results in the finite-dimensional case for both the ${\ell }_1$-norm and the ${\ell }_{\infty }$-norm, we study meteorological data from stations located on the province of Cádiz (Spain). For it, we consider a matrix database with the highest temperature deviations of these stations.