Additivity and multiplicativity come from pairs of approximants. Let \(L\) have rank at most \(n\) and \(K\) rank at most \(m\). If \(L\) approximates \(S\) and \(K\) approximates \(T\), then \(L + K\) has rank at most \(n + m\) and \(\| (S + T) - (L + K)\| \le \| S - L\| + \| T - K\| \). If \(L\) approximates \(R\) and \(K\) approximates \(T\), then \(L T + (R - L) K\) has rank at most \(n + m\) and \(R T - L T - (R - L) K = (R - L)(T - K)\), so \(a_{n+m}(R T) \le \| R - L\| \, \| T - K\| \). Taking the infimum over \(L\) and \(K\) gives both inequalities.

Monotonicity and nonnegativity are inherited from the underlying set of approximants. (S2) and (S3) are the cases of additivity and multiplicativity with one index \(0\), since \(a_0 = \| \cdot \| \), and (S4) is witnessed by the approximant \(A = S\). For (S5\('\)) let \(L\) have \(\operatorname {rank}(L) \le n {\lt} \dim X\); then \(\operatorname {range}(L)\) is a proper finite-dimensional subspace of \(X\), closed because \(\mathbb {K}\) is complete, and Riesz’s lemma provides, for every \(r {\lt} 1\), a vector \(x_0 \ne 0\) with \(\| x_0 - L x_0\| \ge r \| x_0\| \), so \(\| \mathrm{id} - L\| \ge r\); letting \(r \to 1\) and combining with \(a_n(\mathrm{id}) \le \| \mathrm{id}\| = 1\) gives \(a_n(\mathrm{id}_X) = 1\). The classical (S5) on \(\ell _2^{n+1}\) is the special case \(X = \ell _2^{n+1}\).