If \(S(B_X)\) is totally bounded, a finite \(\varepsilon \)-net with \(k\) points bounds \(d_k(S)\) and \(c_k(S)\) by Theorem 11.1, and antitonicity gives the decay.

Conversely, suppose \(d_n(S) {\lt} \varepsilon \) and pick \(V\) of dimension at most \(n\) with \(\| \pi _V \circ S\| {\lt} \varepsilon \). Every \(Sx\) with \(\| x\| \le 1\) is then within \(\varepsilon \) of a point of \(V\) of norm at most \(\| S\| + \varepsilon \); that bounded piece of the finite-dimensional space \(V\) is compact, so a finite \(\varepsilon \)-net of it yields a finite \(2\varepsilon \)-net of \(S(B_X)\).

For \(c_n(S) {\lt} \varepsilon \) pick a closed \(M\) of finite codimension with \(\| S|_M\| {\lt} \varepsilon \). The image of \(B_X\) in the finite-dimensional quotient \(X / M\) is bounded, hence totally bounded; choose a finite \(\delta \)-net inside that image and lift its centres to \(u_1, \dots , u_N \in B_X\). If \(\| [x] - [u_j]\| {\lt} \delta \) there is \(m \in M\) with \(\| (x - u_j) - m\| {\lt} \delta \) and — this is the point — \(\| m\| \le 2 + \delta \), a bound independent of the codimension. Hence

which is small for suitable \(\varepsilon \) and \(\delta \). (Splitting \(x\) by a projection with kernel \(M\) instead would cost the factor \(1 + \sqrt{n}\) of Garling–Gordon (Theorem 7.4), and \(\sqrt{n}\, c_n(S)\) need not tend to \(0\).)

On Hilbert spaces \(s_n(S) = a_n(S)\) by the coincidence theorem (Theorem 4.1), and \(a_n(S) \to 0\) is equivalent to compactness by Theorem 6.10.