2 Axioms of an s-number sequence
A family
\[ s \; =\; (s_n)_{n \in \mathbb {N}_0}, \qquad s_n : \mathcal{L}(X, Y) \longrightarrow [0, \infty ), \]
defined for all pairs of Banach spaces \((X, Y)\) over \(\mathbb {K}\) is called an s-number sequence if:
- (S1)
\(\| S\| = s_0(S) \ge s_1(S) \ge \dots \ge 0\) for every \(S\).
- (S2)
\(s_n(S + T) \le s_n(S) + \| T\| \).
- (S3)
\(s_n(BSA) \le \| B\| \, s_n(S) \, \| A\| \) for composable bounded operators \(A, B\).
- (S4)
\(s_n(S) = 0\) whenever \(\operatorname {rank}(S) \le n\).
- (S5)
\(s_n(\mathrm{id}_{\ell _2^{n+1}}) = 1\).
An s-number sequence is strict if it additionally satisfies the strengthening
- (S5\('\))
\(s_n(\mathrm{id}_X) = 1\) for every Banach space \(X\) with \(\dim X {\gt} n\), not merely on \(X = \ell _2^{n+1}\).