10 Entropy numbers

The entropy numbers quantify the compactness of an operator directly: they measure how many balls of a given radius are needed to cover the image of the unit ball. They are not s-numbers — the rank axiom (S4) and the normalisation (S5) both fail — but they satisfy the remaining properties, in the strong additive and multiplicative form of Theorems 10.3 and 10.4; such sequences are called pseudo-s-number sequences [ 7 , 12.1.1 ] . Everything in this chapter is formalised in Entropy.lean, over a non-trivially normed field $\mathbb {K}$ unless stated otherwise.

Definition 10.1 Entropy number

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For $S \in \mathcal{L}(X, Y)$ and $n \in \mathbb {N}_0$,

\[ e_n(S) := \inf \bigl\{ \varepsilon {\gt} 0 : S(B_X) \text{ is covered by at most } 2^n \text{ closed $\varepsilon $-balls in } Y \bigr\} , \]

where $B_X$ is the closed unit ball of $X$ and the centres of the covering balls are arbitrary points of $Y$. (The indexing is $0$-based, so $e_0$ allows a single ball.)

Proposition 10.2

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For every $S \in \mathcal{L}(X, Y)$ and $n \in \mathbb {N}_0$,

\[ 0 \; \le \; e_{n+1}(S) \; \le \; e_n(S) \; \le \; \| S\| . \]

Proof ▼

Nonnegativity and monotonicity are immediate from the definition (any cover by $2^n$ balls is a cover by $2^{n+1}$ balls). For the norm bound, a single ball of radius $\| S\| + \delta $ centred at the origin covers $S(B_X)$, for every $\delta {\gt} 0$.

Theorem 10.3

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For all $S, T \in \mathcal{L}(X, Y)$ and $n, m \in \mathbb {N}_0$,

\[ e_{n+m}(S + T) \; \le \; e_n(S) + e_m(T) . \]

Proof ▼

If $S(B_X)$ is covered by $2^n$ balls of radius $\varepsilon $ with centres $y_i$ and $T(B_X)$ by $2^m$ balls of radius $\delta $ with centres $z_j$, then the $2^{n+m}$ balls of radius $\varepsilon + \delta $ centred at the sums $y_i + z_j$ cover $(S+T)(B_X)$.

Taking $m = 0$ and using $e_0(T) \le \| T\| $ (Proposition 10.2) recovers the s-number axiom (S2): $e_n(S + T) \le e_n(S) + \| T\| $.

Theorem 10.4

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Let the norm values of $\mathbb {K}$ be dense in $[0, \infty )$ (as for $\mathbb {R}$ and $\mathbb {C}$). Then for all $S \in \mathcal{L}(X, Y)$, $B \in \mathcal{L}(Y, Z)$ and $n, m \in \mathbb {N}_0$,

\[ e_{n+m}(B \circ S) \; \le \; e_n(B)\, e_m(S) . \]

Proof ▼

Cover $S(B_X)$ by $2^m$ balls of radius $\delta $ with centres $y_i$. Each point of $S(B_X)$ in the ball around $y_i$ differs from $y_i$ by a vector $u$ with $\| u\| \le \delta $, which is rescaled as $u = c\, w$ with $\| w\| \le 1$ and $|c|$ close to $\delta $ — this is where density of the norm values enters. Covering $B(B_Y)$ by $2^n$ balls of radius $\varepsilon $ then covers each $B(u)$ by $2^n$ balls of radius $|c|\, \varepsilon $, and combining the two covers yields $2^{n+m}$ balls of radius close to $\varepsilon \delta $ around $B(S(B_X))$.

Taking one index to be $0$ and using $e_0 \le \| \cdot \| $ twice (Proposition 10.2) gives the s-number axiom (S3): $e_n(B \circ S \circ A) \le \| B\| \, e_n(S) \, \| A\| $.

Remark 10.5 Entropy numbers are not s-numbers

Two of Pietsch’s axioms fail, and one is not formalised:

(S4) fails: if $e_n(S) = 0$ then $S(B_X)$ is covered by $2^n$ balls of arbitrarily small radius, hence has at most $2^n$ points — impossible for $S \neq 0$. So $e_n(S) {\gt} 0$ even for finite-rank $S$.
(S5) fails for $n \ge 2$: covering the Euclidean unit ball of $\mathbb {R}^{n+1}$ by balls of radius ${\lt} 1$ takes only $n + 2$ of them, and $2^n \ge n + 2$ from $n = 2$ on, so $e_n(\mathrm{id}_{\ell _2^{n+1}}) {\lt} 1$.
(S1a), the equality $e_0(S) = \| S\| $, holds over $\mathbb {R}$ and $\mathbb {C}$ by symmetry of $S(B_X)$, but fails over general normed fields; only the inequality $e_0(S) \le \| S\| $ is formalised, which is all that is used anywhere in this blueprint.

Theorem 10.6

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For every $S \in \mathcal{L}(X, Y)$, the sequence $e_n(S)$ tends to $0$ if and only if $S(B_X)$ is totally bounded. If $Y$ is complete, then

\[ S \text{ is compact} \iff e_n(S) \xrightarrow {\, n \to \infty \, } 0 , \]

where the forward implication needs no completeness.

Proof ▼

The first equivalence matches the definitions: a finite $\varepsilon $-net of $S(B_X)$ of size $N$ is a cover by $2^n$ balls once $2^n \ge N$, and conversely. If $S$ is compact, the closure of $S(B_X)$ is compact, hence totally bounded; if conversely $S(B_X)$ is totally bounded and $Y$ is complete, its closure is compact, which is the definition of a compact operator.