Vector and operator norms

Assuming familiarity with vector spaces, we begin by recapitulating the properties of norms and inner products for the vector space \( \mathbf{\textit{V}} = \mathbb{C}^{\textit{d}} \) , which will serve as the framework for developing a theory of logarithmic norms of matrices in \( \mathbb{C}^{\textit{d}\times d} \) . We also collect the properties of operator norms, emphasizing least upper bounds as well as greatest lower bounds.