8 Examples

This chapter computes the s-numbers of three concrete families of operators. The finite-dimensional examples — the identity between \(\ell _p\)-spaces and the diagonal operators — are the standard test cases of the theory; the inclusion \(\ell _1 \to \ell _\infty \) shows that the linear factor \(n+1\) in the maximal difference theorem (Theorem 7.13) cannot be improved in order. The chapter is a leaf of the development: it consumes the general theory and nothing depends on it. Throughout, \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \), and \(\ell _p^m\) denotes \(\mathbb {K}^m\) with the \(\ell _p\)-norm.