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Some Classes of Operators

  • Bernard Aupetit
Chapter
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Part of the Universitext book series (UTX)

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

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Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • Bernard Aupetit
    • 1
  1. 1.Départment de Mathématiques et StatistiqueUniversité LavalQuébecCanada

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