3.4.1 Pietsch’s lifting identity
For a Banach space \(X\) let \(Q_X : \ell ^1(B_X) \to X\) be the summation map on the \(\ell ^1\) space indexed by the closed unit ball \(B_X\),
The numbers \(d^Q_n(S) := a_n(S \circ Q_X)\) give a second formalisation of the Kolmogorov numbers, in KolmogorovLifting.lean. They are defined only for operators on Banach spaces, so they are no s-number sequence in the sense of Definition 2.1; on Banach spaces they satisfy the axioms one by one, and Pietsch’s identity (Theorem 3.10) shows \(d^Q_n = d_n\).
Let \(\mathbb {K}\) be a complete densely normed field. For operators whose domains are Banach spaces, \(d^Q\) satisfies (S1)–(S4), and \(d^Q_n(\mathrm{id}_X) = 1\) for every Banach space \(X\) with \(\dim X {\gt} n\).
Each axiom except the ideal property reduces to the corresponding fact for the approximation numbers (Theorem 3.2); the value \(d^Q_0(S) = \| S \circ Q_X\| = \| S\| \) uses that every \(x \in B_X\) is \(Q_X(\delta _x)\) and that the norm values of \(\mathbb {K}\) are dense. The ideal property (S3) factors \((B \circ S \circ A) \circ Q_W = B \circ (S \circ Q_X) \circ T\) through a lift \(T : \ell ^1(B_W) \to \ell ^1(B_X)\) that sends the basis vector \(\delta _w\) to \(c \cdot \delta _{c^{-1} A w}\) for a scalar \(c\) with \(\| A\| {\lt} |c|\), so that \(\| T\| \le |c|\); letting \(|c| \searrow \| A\| \), using that the norm values of \(\mathbb {K}\) are dense, yields the bound \(\| B\| \, d^Q_n(S) \, \| A\| \). Nonnegativity, monotonicity, (S2) and (S4) hold over every non-trivially normed field.
Let \(\mathbb {K}\) be a complete densely normed field. For every Banach space \(X\) and every \(S \in \mathcal{L}(X, Y)\), the canonical Kolmogorov number equals the approximation number of the lift: \(d_n(S) = a_n(S \circ Q_X)\). Completeness of \(\mathbb {K}\) is needed: over \(\mathbb {Q}\), with \(X = Y = \mathbb {R}\) and \(S = \mathrm{id}\), one has \(d_1(S) = 0\) because \(\mathbb {Q}\) is dense in \(\mathbb {R}\), while \(a_1(Q_X) = 1\).
The inequality \(d_n(S) \le a_n(S \circ Q_X)\): for a rank-\(\le n\) approximant \(L\) of \(S \circ Q_X\), take \(V := \operatorname {range} L\); since every unit vector \(x\) is \(Q_X(\delta _x)\) and \(\pi _V \circ L = 0\), one gets \(\| \pi _V \circ S\| \le \| S \circ Q_X - L\| \). The reverse inequality: for an admissible \(V\) and \(\varepsilon {\gt} 0\), pick \(w_x \in V\) with \(\| S x - w_x\| {\lt} \| \pi _V \circ S\| + \varepsilon \) and lift to \(L(\alpha ) := \sum _x \alpha (x)\, w_x\) (a rank-\(\le n\) operator into the finite-dimensional \(V\)), so that \(\| S \circ Q_X - L\| \le \| \pi _V \circ S\| + \varepsilon \).