We investigate different possiblities of subspaces of the space $\ell_{ \infty}$ in terms of whether the subspaces are polyhedral or not. We further study finite-dimensional subspaces of $\ell_{ \infty}$ which are of the form $\ell_\infty^n$ form some $ n \geq 2.$ As an application of the results we compute the number of extreme contractions for a class of the space of bounded linear operators. In particular we find the number of extreme contractions of $\mathbb{ L}(\mathbb{ X}, \ell_{ \infty}^n),$ where $\mathbb{ X}$ is a finite-dimensional polyhedral space.
Categories: math.FA