11.3 Entropy bound by Gelfand and Kolmogorov numbers
If \(S(B_X)\) contains more than \(2^n\) points that are pairwise at distance at least \(\delta \), then \(e_n(S) \ge \delta / 2\).
A ball of radius \(\varepsilon \) has diameter at most \(2\varepsilon \), so if \(2\varepsilon {\lt} \delta \) then each ball of an admissible cover contains at most one of the points. Assigning to every point a covering centre is then injective, so there are at least as many centres as points, more than \(2^n\), contradicting admissibility.
Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \). For every \(S \in \mathcal{L}(X, Y)\) with \(S \neq 0\) and every \(n \in \mathbb {N}_0\), \(e_n(S) {\gt} 0\). In particular (S4) fails for every nonzero operator of finite rank.
Take a unit vector \(x\) with \(v = Sx \neq 0\). The \(2^n + 1\) points \(\frac{k}{2^n} v\), \(0 \le k \le 2^n\), are images of the vectors \(\frac{k}{2^n} x \in B_X\) and lie at mutual distance at least \(\| v\| / 2^n\), so \(e_n(S) \ge \| v\| / 2^{n+1}\) by Proposition 11.5.
Let \(\gamma {\lt} d_n(S)\). Then there are \(x_0, \dots , x_n \in B_X\) and subspaces \(V_0, \dots , V_n \subseteq Y\) with \(S x_k \in V_j\) for \(k {\lt} j\) and \(\| S x_j + w\| {\gt} \gamma \) for all \(w \in V_j\). Dually, for \(0 {\lt} \gamma {\lt} c_n(S)\) there are \(x_0, \dots , x_n \in B_X\) and functionals \(\rho _0, \dots , \rho _n\) with \(\| \rho _j\| \le 1\), \(\rho _j(S x_j) = \gamma \) and \(\rho _i(S x_j) = 0\) for \(i {\lt} j\).
Both are built one vector at a time. For \(d_n\), take \(V_j\) to be the span of the images already chosen (of dimension at most \(n\)) and pick \(x_j \in B_X\) with \(\| S x_j + w\| {\gt} \gamma \) for all \(w \in V_j\), which is possible because \(\gamma {\lt} d_n(S)\) and \(\dim V_j \le n\) (Definition 3.7). For \(c_n\), make the same selection below \(c_n(S)\) (using \(\gamma {\lt} c_n(S)\) and Definition 3.5) on the common kernel of the functionals already chosen, a closed subspace of codimension at most \(n\), and rescale the resulting unit-image vector by \(\gamma \) to land in \(B_X\).
For every \(S \in \mathcal{L}(X, Y)\) and \(n \in \mathbb {N}_0\),
Let \(\gamma {\lt} d_n(S)\) and take a flag \(x_0, \dots , x_n\) as in Theorem 11.7. For \(\sigma \in \{ \pm 1\} ^{n+1}\) set
For \(\sigma \neq \tau \) let \(j\) be the largest index with \(\sigma _j \neq \tau _j\). Then \(z_\sigma - z_\tau = \tfrac {1}{n+1}\bigl( (\sigma _j - \tau _j) S x_j + w \bigr)\) with \(w \in V_j\), because the indices \(k {\gt} j\) contribute nothing and those \(k {\lt} j\) contribute elements of \(V_j\). Since \(|\sigma _j - \tau _j| = 2\), the flag property gives \(\| z_\sigma - z_\tau \| {\gt} 2\gamma /(n+1)\). So \(2^{n+1}\) points of \(S(B_X)\) are pairwise more than \(2\gamma /(n+1)\) apart, and Proposition 11.5 yields \(e_n(S) \ge \gamma /(n+1)\). Letting \(\gamma \nearrow d_n(S)\) gives the claim.
For \(c_n\) the same computation is read off by the functional \(\rho _j\) at the smallest disagreeing index \(j\): it annihilates \(S x_k\) for \(k {\gt} j\), while \(\sigma _k = \tau _k\) kills the terms with \(k {\lt} j\).