Call a closed subspace of codimension at most \(n\) admissible at stage \(n\). Codimensions add up under intersections, \(\operatorname {codim}(M \cap N) \le \operatorname {codim} M + \operatorname {codim} N\), since the map \(X \to X/M \times X/N\), \(x \mapsto ([x]_M, [x]_N)\), has kernel \(M \cap N\) (rank_quotient_inf_le). For additivity, let \(M, N \subseteq X\) be admissible at stages \(n\) and \(m\). Then \(M \cap N\) is closed and admissible at stage \(n + m\), and \(\| (S + T)\restriction _{M \cap N}\| \le \| S \restriction _M\| + \| T \restriction _N\| \). For multiplicativity, let \(M \subseteq Y\) and \(N \subseteq X\) be admissible at stages \(n\) and \(m\). Then \(T^{-1}(M) \cap N\) is closed and admissible at stage \(n + m\), since \(\operatorname {codim} T^{-1}(M) \le \operatorname {codim} M\), and for \(x\) in it \(\| R T x\| \le \| R \restriction _M\| \, \| T x\| \le \| R \restriction _M\| \, \| T \restriction _N\| \, \| x\| \). Taking the infimum over \(M\) and \(N\) gives both inequalities.

The axioms are dual to the Kolmogorov case (Theorem 3.8) under the domain \(\leftrightarrow \) codomain swap: where the Kolmogorov numbers push \(S\) down a quotient projection \(\pi _V : Y \to Y/V\) for a small \(V \subseteq Y\), the Gelfand numbers restrict \(S\) along the inclusion \(\iota _M : M \hookrightarrow X\) for a subspace \(M \subseteq X\) of small codimension. With \(M = X\) one gets \(c_0(S) = \| S\| \). (S2) and (S3) are the cases of additivity and multiplicativity with one index \(0\). For (S4) take \(M := \ker S\), which is closed with \(\operatorname {codim} M = \operatorname {rank} S \le n\) and \(S \circ \iota _M = 0\). For (S5\('\)), the inclusion \(\iota _M\) is an isometry, so the identity restricted to any nontrivial \(M\) has norm \(1\), and \(\operatorname {codim} M \le n {\lt} \dim X\) forces \(M \neq \{ 0\} \).