9 Injective and surjective s-numbers
Two further structural properties of an s-number sequence, from Pietsch’s theory, single out the Gelfand and Kolmogorov numbers. They concern the behaviour of \(s_n\) under composition with two distinguished classes of maps.
A metric injection is a continuous linear map \(J : Y \to Z\) that preserves norms: \(\| J y\| = \| y\| \) for all \(y\) (an isometric embedding). In particular \(\| J\| \le 1\).
A metric surjection is a continuous linear map \(Q : W \to X\) that is a contraction (\(\| Q\| \le 1\)) and maps the open unit ball onto the open unit ball: every \(x\) has preimages of norm arbitrarily close to \(\| x\| \). This is the quotient-map condition \(\| x\| = \inf \{ \| w\| : Q w = x\} \).
Post-composition with a metric injection and pre-composition with a metric surjection both leave the operator norm unchanged: \(\| J \circ T\| = \| T\| \) and \(\| T \circ Q\| = \| T\| \).
For the injection, \(\| (J \circ T)x\| = \| J(Tx)\| = \| Tx\| \) pointwise, so the two operator norms agree. For the surjection, \(\| T \circ Q\| \le \| T\| \) since \(\| Q\| \le 1\); conversely, lifting each \(x\) to a preimage \(w\) with \(\| w\| \approx \| x\| \) gives \(\| Tx\| = \| (T\circ Q)w\| \le \| T\circ Q\| \, \| w\| \), and letting the slack tend to \(0\) yields \(\| T\| \le \| T \circ Q\| \).
A family \(s\) as in Definition 2.1 is injective if post-composition with any metric injection leaves it unchanged, \(s_n(J \circ S) = s_n(S)\); it is surjective if pre-composition with any metric surjection leaves it unchanged, \(s_n(S \circ Q) = s_n(S)\).
The Gelfand numbers \(c_n\) form an injective s-number sequence: \(c_n(J \circ S) = c_n(S)\) for every metric injection \(J\).
The infimum defining \(c_n\) ranges over restriction norms \(\| S|_M\| \), and \(\| (J \circ S)|_M\| = \| J \circ (S|_M)\| = \| S|_M\| \) by Lemma 9.3. The two infimum sets therefore coincide term by term, so the Gelfand numbers agree.
The Kolmogorov numbers are surjective: \(d_n(S \circ Q) = d_n(S)\) for every metric surjection \(Q\).
The infimum defining \(d_n\) ranges over quotient norms \(\| \pi _V \circ S\| \), and \(\| \pi _V \circ (S \circ Q)\| = \| (\pi _V \circ S) \circ Q\| = \| \pi _V \circ S\| \) by Lemma 9.3. The two infimum sets coincide term by term, so the Kolmogorov numbers agree.
On Banach spaces over a complete densely normed field, the Kolmogorov numbers are the approximation numbers after the summation map \(Q_X : \ell _1(B_X) \to X\) (Theorem 3.10). Dually, the Gelfand numbers are the approximation numbers after a metric injection of the codomain. Let \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \) and let \(J_Y : Y \to \ell _\infty (B_{Y^*})\), \((J_Y y)(\varphi ) = \varphi (y)\), be the canonical embedding into the bounded families indexed by the closed unit ball of the dual.
\(J_Y\) is a metric injection: \(\| J_Y y\| = \| y\| \) for every \(y \in Y\).
Every coordinate is a functional of norm at most \(1\), so \(\| J_Y y\| \le \| y\| \), and a norming functional of \(y\) (Hahn–Banach) gives \(\| J_Y y\| \ge \| y\| \).
For all normed spaces \(X\), \(Y\) over \(\mathbb {K}\), every \(S \in \mathcal{L}(X, Y)\) and every \(n \in \mathbb {N}_0\),
By injectivity (Theorem 9.5) and the sandwich theorem, \(c_n(S) = c_n(J_Y S) \le a_n(J_Y S)\). Conversely, let \(M \subseteq X\) be closed of codimension at most \(n\). Extend each functional \((\varphi \circ S)|_M\), \(\varphi \in B_{Y^*}\), to \(g_\varphi \in X^*\) of the same norm (Hahn–Banach), so \(\| g_\varphi \| \le \| S|_M\| \). The operator \(L x = (\varphi (S x) - g_\varphi (x))_\varphi \) vanishes on \(M\), hence factors through \(X/M\) and has rank at most \(n\), and \((J_Y S - L)x = (g_\varphi (x))_\varphi \) has norm at most \(\| S|_M\| \, \| x\| \). Thus \(a_n(J_Y S) \le \| S|_M\| \), and the infimum over \(M\) gives \(a_n(J_Y S) \le c_n(S)\). The proof uses neither completeness of \(X\) nor of \(Y\), in contrast to the lifting identity.