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. Everything in this chapter is fully formalised in Injectivity.lean.
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 \(\| T\| \le \| T \circ Q\| \).
An s-number sequence \(s\) 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 \(d_n\) form a surjective s-number sequence: \(d_n(S \circ Q) = d_n(S)\) 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.