3.4 Kolmogorov numbers \(d_n\)
For \(S \in \mathcal{L}(X, Y)\) and \(n \in \mathbb {N}_0\),
Let \(\mathbb {K}\) be a non-trivially normed field and \(X, Y\) normed spaces over \(\mathbb {K}\). Then the Kolmogorov numbers \(d = (d_n)_n\) form a strict s-number sequence: they satisfy (S1)–(S5) and the strengthening (S5\('\)).
The proof works with the operator-norm-on-quotient identity \(d_n(S) = \inf _V \| \pi _V \circ S\| \), where \(\pi _V : Y \to Y/V\) is the canonical projection onto the quotient. (S2) is the triangle inequality applied to \(\pi _V \circ S\); for (S3), an admissible \(V\) for \(S\) transports to the admissible subspace \(B(V)\) for \(BSA\), with the deviation bounded per point; for (S4) take \(V = \operatorname {range} S\). For (S5\('\)), given a subspace \(V\) of dimension \(\le n {\lt} \dim X\), its closure is proper and Riesz’s lemma produces unit vectors almost at distance \(1\) from \(V\), so \(\| \pi _V\| = 1\) and \(d_n(\mathrm{id}_X) = 1\).
The (S5\('\)) step additionally needs \(\mathbb {K}\) complete (to close finite-dimensional subspaces via Riesz’s lemma); it requires neither a densely normed \(\mathbb {K}\) nor completeness of \(X\).