2 Axioms of an s-number sequence
A family
defined for all pairs of normed 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 normed space \(X\) with \(\dim X {\gt} n\), not merely on \(X = \ell _2^{n+1}\).
Two further classical strengthenings concern the axioms (S2) and (S3); see [ 7 ] and [ 5 , Section 5.1 ] .
A family \(s = (s_n)_{n \in \mathbb {N}_0}\) as in Definition 2.1 is additive if for all normed spaces \(X, Y\), all \(S, T \in \mathcal{L}(X, Y)\) and all \(n, m \in \mathbb {N}_0\),
A family \(s = (s_n)_{n \in \mathbb {N}_0}\) as in Definition 2.1 is multiplicative if for all normed spaces \(X, Y, Z\), all \(T \in \mathcal{L}(X, Y)\), \(S \in \mathcal{L}(Y, Z)\) and all \(n, m \in \mathbb {N}_0\),
Together with \(s_0 = \| \cdot \| \), additivity implies (S2), by taking \(m = 0\), and multiplicativity implies (S3), by two applications with one index \(0\): \(s_n(BSA) \le s_0(B) \, s_n(SA) \le \| B\| \, s_n(S) \, s_0(A)\) (Additive.subadditive, Multiplicative.idealLike). The approximation, Gelfand and Kolmogorov numbers are additive and multiplicative and obtain (S2) and (S3) in this way; the Hilbert numbers are additive, and the Bernstein numbers are neither (Theorem 3.4).
For every s-number sequence \(s\), all \(S, T \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),
By (S2), \(s_n(S) = s_n(T + (S - T)) \le s_n(T) + \| S - T\| \), and symmetrically \(s_n(T) \le s_n(S) + \| S - T\| \).