11.5 All s-numbers
Let \(s\) be an s-number sequence. For every \(S \in \mathcal{L}(X, Y)\) and \(n \in \mathbb {N}_0\),
\[ s_n(S) \; \le \; 2e\, (n+1)\, e_n(S) . \]
Proof
No bound \(a_n(S) \le C\, e_n(S)\) with a constant \(C\) holds for all operators between Banach spaces: compact operators into a Banach space have \(e_n(S) \to 0\) (Theorem 10.7), and a bound of this form would make every compact operator a norm limit of finite-rank operators, which fails on Banach spaces without the approximation property (Enflo). For the Gelfand and Kolmogorov numbers the factor is \(n+1\) (Theorem 11.8), for the Hilbert numbers \(2\) (Theorem 11.10).