Write \(\gamma (S, M) := \inf _{0 \ne x \in M} \| Sx\| /\| x\| \) for the gain of \(S\) on a subspace \(M\), so that \(b_n(S) = \sup _{\dim M = n+1} \gamma (S, M)\). On a one-dimensional \(M = \operatorname {span}\{ v\} \) the gain is exactly \(\| Sv\| /\| v\| \), and the supremum over \(v\) gives \(b_0(S) = \| S\| \); monotonicity in \(n\) holds because every \((n+2)\)-dimensional subspace contains an \((n+1)\)-dimensional one, and shrinking \(M\) only enlarges the gain. Subadditivity (S2) is the pointwise triangle inequality \(\| (S+T)x\| /\| x\| \le \| Sx\| /\| x\| + \| T\| \), pushed through the infimum and the supremum. For the ideal property (S3) fix an \((n+1)\)-dimensional \(M \subseteq W\): if \(A\) is not injective on \(M\), the gain of \(BSA\) on \(M\) vanishes; if it is, then \(M' := A(M)\) has dimension \(n+1\) and \(\gamma (BSA, M) \le \| B\| \, \gamma (S, M') \, \| A\| \le \| B\| \, b_n(S) \, \| A\| \). For the rank axiom (S4), an operator of rank \(\le n\) has nontrivial kernel on every \((n+1)\)-dimensional subspace by rank–nullity, forcing the gain to zero. Finally (S5\('\)): the identity has gain \(1\) on every nontrivial subspace, and \(\dim X {\gt} n\) guarantees an \((n+1)\)-dimensional subspace exists. All of this needs only a non-trivially normed scalar field: no completeness, and no Riesz lemma.

For the two failures let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \) and write \(\ell _p^N\) for \(\mathbb {K}^N\) with the \(p\)-norm. Two bounds on the identities enter:

For the first bound (bernsteinNumber_idEmbed_two_infty_le) let \(M \subseteq \mathbb {K}^N\) have dimension \(n + 1\). By the flatness lemma behind the Gelfand-width lower bounds, an extreme point argument, \(M\) contains a vector \(x\) with \(|x_i| \le 1\) for all \(i\) and \(|x_i| = 1\) for at least \(n + 1\) indices \(i\); then \(\| x\| _\infty \le 1\) and \(\| x\| _2 \ge \sqrt{n+1}\). For the second bound (bernsteinNumber_idEmbed_infty_two_le) let \(c\) be the gain of the identity \(\ell _\infty ^N \to \ell _2^N\) on an \((n+1)\)-dimensional \(M\), and let \(v_1, \dots , v_{n+1}\) be an orthonormal basis of \(M\) for the inner product of \(\ell _2^N\). For a coordinate \(k\) put \(s_k := \sum _i |v_i(k)|^2\) and \(x := \sum _i \overline{v_i(k)} \, v_i \in M\). Then \(x(k) = s_k\) and \(\| x\| _2^2 = s_k\), so \(c \, s_k \le c \, \| x\| _\infty \le \| x\| _2 = \sqrt{s_k}\), that is, \(c^2 s_k \le 1\); summing over \(k\) and using \(\sum _k s_k = \sum _i \| v_i\| _2^2 = n + 1\) gives \(c^2 (n+1) \le N\).

Multiplicativity fails for \(T = \mathrm{id} : \ell _2^3 \to \ell _\infty ^3\) and \(S = \mathrm{id} : \ell _\infty ^3 \to \ell _2^3\) (bernsteinNumber_mul_lt_comp): the composition \(S T\) is the identity of \(\ell _2^3\), so \(b_2(S T) = 1\) by (S5\('\)), while the two bounds give \(b_1(S) \, b_1(T) \le \sqrt{3/2} \cdot 1/\sqrt{2} = \sqrt{3}/2 {\lt} 1\). Additivity fails on \(Z = \ell _2^{71} \times \ell _\infty ^{71}\) with the norm \(\| (x, y)\| = \max \{ \| x\| _2, \| y\| _\infty \} \) for the operators \(S(x, y) = (0, x)\) and \(T(x, y) = (y/9, 0)\) (bernsteinNumber_add_lt): on the \(71\)-dimensional subspace \(E = \{ (x, 3x) : x \in \mathbb {K}^{71}\} \) the operator \(S + T\) acts as multiplication by \(1/3\), since \((S + T)(x, 3x) = (x/3, x)\), hence \(b_{70}(S + T) \ge 1/3\); by the ideal property and the two bounds, \(b_{35}(S) \le b_{35}(\mathrm{id} : \ell _2^{71} \to \ell _\infty ^{71}) \le 1/6\) and \(b_{35}(T) \le \tfrac 19 \, b_{35}(\mathrm{id} : \ell _\infty ^{71} \to \ell _2^{71}) \le \tfrac 19 \sqrt{71/36} {\lt} \tfrac 16\), since \(71 {\lt} 81\).