Some Classes of Operators

Abstract

Let X be a finite-dimensional vector space. We know that all norms on X are equivalent and this implies in particular that all linear mappings T from X into X are continuous. Indeed if ‖·‖ is a norm on X and if e₁,...,e _n is a basis of X, then we have

$$\begin{array}{*{20}{c}} {\parallel Tx\parallel \leqslant (|{{\lambda }_{1}}| + \cdots + |{{\lambda }_{n}}|)\mathop{{\max }}\limits_{{i = 1, \ldots ,n}} \parallel T{{e}_{i}}\parallel ,} & {for x = {{\lambda }_{1}}{{e}_{1}} + \cdots + {{\lambda }_{n}}{{e}_{n}}.} \\ \end{array}$$