11.1 Covering estimates
Let \(S \in \mathcal{L}(X, Y)\) and suppose every \(x \in B_X\) satisfies \(\| Sx - y_i\| \le \varepsilon \) for some member of a family \(y_1, \dots , y_k \in Y\). Then
For \(d_k\) take \(V := \operatorname {span}\{ y_1, \dots , y_k\} \), of dimension at most \(k\). Since \(y_i \in V\), the quotient map satisfies \(\| \pi _V(Sx)\| = \| \pi _V(Sx - y_i)\| \le \| Sx - y_i\| \le \varepsilon \), so \(\| \pi _V \circ S\| \le \varepsilon \).
For \(c_k\) choose norming functionals \(b_i\) with \(\| b_i\| \le 1\) and \(b_i(y_i) = \| y_i\| \) (Hahn–Banach) and let \(M\) be the common kernel of the \(b_i \circ S\), a closed subspace of codimension at most \(k\). For \(x \in M\) with \(\| x\| \le 1\) pick \(i\) with \(\| Sx - y_i\| \le \varepsilon \); then \(\| y_i\| = b_i(y_i - Sx) \le \varepsilon \) and hence \(\| Sx\| \le \| Sx - y_i\| + \| y_i\| \le 2\varepsilon \).
For every \(S \in \mathcal{L}(X, Y)\) and \(n \in \mathbb {N}_0\),
An admissible radius \(\varepsilon \) for \(e_n(S)\) provides a net of at most \(2^n\) points, so Theorem 11.1 applies with \(k \le 2^n\); both sequences are antitone in the index.