8.1 The identity \(\mathrm{id} : \ell _q^m \to \ell _p^m\)

Theorem 8.1

✓

For \(1 \le p \le q {\lt} \infty \) and \(m \ge 1\),

\[ \bigl\| \mathrm{id} : \ell _q^m \to \ell _p^m \bigr\| \; =\; m^{\, 1/p - 1/q}. \]

Proof ▼

The upper bound is the power-mean (Hölder) inequality \(\| x\| _p \le m^{1/p-1/q} \| x\| _q\); the lower bound tests the all-ones vector, whose \(\ell _q\)-norm is \(m^{1/q}\) and whose \(\ell _p\)-norm is \(m^{1/p}\).

Theorem 8.2

✓

For \(1 \le p \le q {\lt} \infty \) and \(n {\lt} m\),

\[ a_n\bigl( \mathrm{id} : \ell _q^m \to \ell _p^m \bigr) \; =\; (m - n)^{\, 1/p - 1/q}. \]

Proof ▼

For the upper bound, truncate to the first \(n\) coordinates: the residual is the identity on the remaining \(m - n\) coordinates, of norm \((m-n)^{1/p-1/q}\) by (the proof of) Theorem 8.1. For the lower bound, a rank-\(\le n\) operator \(L\) has kernel of dimension at least \(m - n\), and a flatness (pigeonhole) argument produces a nonzero \(x\) in the kernel with \(\| x\| _p \ge (m-n)^{1/p-1/q} \| x\| _q\); on such an \(x\) the error \(\mathrm{id} - L\) acts as the identity, so \(\| \mathrm{id} - L\| \ge (m-n)^{1/p-1/q}\).

By the sandwich theorem (Theorem 4.2), every s-number sequence obeys \(s_n(\mathrm{id}) \le (m-n)^{1/p-1/q}\).

Proposition 8.3

✓

For every strict s-number sequence \(s\), every \(1 \le p \le \infty \) and \(n {\lt} m\), we have \(s_n(\mathrm{id}_{\ell _p^m}) = 1\).

Proof ▼

This is exactly the strict normalisation (S5\('\)) applied to \(X = \ell _p^m\), whose dimension is \(m {\gt} n\).