B. S. MITYAGIN

THE RELATION BETWEEN ε-ENTROPY, THE RATE OF APPROXIMATION, AND THE NUCLEARITY OF A COMPACT SET IN A LINEAR SPACE

(Presented by Academician A. N. Kolmogorov, 23 IV 1960)

A. N. Kolmogorov ((^{1})) and A. Pelczynski ((^{2})) introduced the concept of the approximative dimension of linear topological spaces. In the present note there is obtained, in particular, in terms of this concept, a complete characterization of nuclear spaces ((^{3})), which are very important in analysis. Our results give a complete solution of a problem of I. M. Gelfand (see ((^{4})), p. 6), posed at the Third All-Union Conference on Functional Analysis (January 1956).

In what follows we shall be concerned with locally convex linear topological spaces, real or complex, although, for simplicity, some computations are given only for the real case.

1. Let (E) be a linear space, (S) a centrally symmetric convex (c.s.c.) set in (E), and (A) an arbitrary set in (E).

Definition 1 (see ((^{1,5,6}))). Put

[
N(A,S;\varepsilon)=\inf_{x_k\in E}\left{N:\bigcup_1^N (x_k+\varepsilon S)\supset A\right},
]

[
M(A,S;\varepsilon)=\sup_{x_k\in A}\left{M:x_k-x_j\notin \varepsilon S;\ j\ne k;\ k,j=1,\ldots,M\right}.
]

The number (H(A,S;\varepsilon)=\log N(A,S;\varepsilon)) will be called the (\varepsilon)-entropy of (A) relative to (S), and the number (\log M(A,S;\varepsilon)) its (\varepsilon)-capacity.

Definition 2 (cf. ((^{6})), p. 22). We shall call the order (\rho(A,S)) of the set (A) relative to the c.s.c. set (S) the number

[
\rho(A,S)=\lim_{\varepsilon\to0}\left{\log H(A,S;\varepsilon)\big/\log \frac{1}{\varepsilon}\right}.
]

Lemma 1 (((^{6})), p. 8).
[
M(A,S;\varepsilon)\ge N(A,S;\varepsilon)\ge M(A,S;2\varepsilon).
]

Lemma 2 ((^{7})). Let (U_i,\ i=1,\ldots,n), be c.s.c. sets in (E) and

[
\rho_{ki}=\rho(U_k,U_i).
]

Then

[
\frac{1}{\rho_{n1}}\ge \sum_1^{n-1}\frac{1}{\rho_{k+1,k}}.
]

Let ({u_1,\ldots,u_n}) be a basis in the (n)-dimensional space (E_n), and let (p_n) be the octahedron generated by the vectors (a_k u_k,\ k=1,\ldots,n) ((a_k>0) and fixed), i.e.

[
p_n=\left{\sum_1^n \xi_k a_k u_k:\sum_1^n |\xi_k|\le 1\right}.
]

Lemma 3. The number of points of the form

[
y=\sum_1^n \gamma_i u_i,
]

where (\gamma_i) are integers, lying in (p_n), is not less than

[
\prod_1^n \frac{a_k}{k}.
]

Indeed, if each point (y=\sum_1^n \gamma_i u_i), (\gamma_i) integers, lying in (p_n), is surrounded by a cube (K_y) with side length “two,” then (\bigcup_{y\in p_n} K_y \supset p_n). Counting the volumes of these sets leads to the assertion of the lemma.

Definition 3 (see ((^8))). The (n)-dimensional width of a set (A) in a Banach space (B) with unit ball (S) is
[
d_n=d_n(A,S)=\inf r(A,E_n),
]
where
[
r(A,E_n)=\sup_{x\in A}\inf_{y\in E_n}|x-y|_S
]
and the infimum is taken over all (n)-dimensional subspaces (E_n) in (B).

1. Computation of the functions (H(\varepsilon)) and (d_n) for concrete compacta in many function spaces shows ((^6)) that the asymptotics of these functions are closely connected. Below this connection is investigated for any centrally symmetric convex compactum (K) in an arbitrary Banach space (B).

Let us consider the functions
[
m(t)=\max\left{n:d_n\geq \frac1t\right}
]
and
[
l(t)=\max\left{n:\frac{d_n}{n}\geq \frac1t\right},
]
which characterize the compactum (K).

Lemma 4 (( (^9), \text{p. }20)).
[
\alpha=\inf\left{\beta:\sum d_n^\beta<\infty\right}
=\overline{\lim}_{t\to 0}{\log m(t)/\log t}.
]

Theorem 1. In an arbitrary Banach space (B), for every centrally symmetric convex compactum (K) the inequalities
[
m\left(\frac{2}{\varepsilon}\right)\log \frac{8(d_0+\varepsilon)}{\varepsilon}
\geq H(K,S;\varepsilon)\geq
\int_0^{1/\varepsilon}\frac{l(t)}{t}\,dt .
\tag{1}
]

The left-hand inequality(^*), valid for any compactum, is easily obtained by using the discussion in § 4 of ((^6)).

The right-hand inequality. Let (a_1=d_0=\sup_{x\in K}|x|); since (K) is compact, for some (x_1\in K) we have (|x_1|=a_1). By induction one constructs a system of vectors (x_i), (i=1,2,\ldots), such that (x_i\in K) and
[
a_{n+1}=r(x_{n+1},L_n)\geq d_n
]
((1)), where (L_n=L{x_1,\ldots,x_n}); this construction is possible, since (see Definition 3) (r(K,L_n)\geq d_n) and (K) is compact.

By the convexity and central symmetry of (K), for all (n) it contains the octahedra
[
\sigma_n=\left{\sum_1^n \xi_k x_k:\sum_1^n |\xi_k|\leq 1\right}.
]
For any({}^{**}) two distinct points
[
y_1=\sum_1^n \alpha_k \varepsilon a_k^{-1}x_k,\qquad
y_2=\sum_1^n \beta_k \varepsilon a_k^{-1}x_k,
]
where (\alpha_k,\beta_k) are integers, we have
[
z=y_1-y_2=\sum_1^l \gamma_k \varepsilon a_k^{-1}x_k,\qquad
1\leq l\leq n,\quad |\gamma_l|\geq 1,
]
and
[
|z|\geq r(z,L_{l-1})=r(\gamma_l\varepsilon a_l^{-1}x_l,L_{l-1})\geq \varepsilon,
]
so that, by Lemma 3,
[
M(\sigma_n,S;\varepsilon)\geq \prod_1^n \frac{a_k}{\varepsilon k}.
]
In view of Lemma 1, inequalities (1), and the inclusions (K\supset\sigma) for all (n), we have
[
N(K,S;\varepsilon)\geq \prod_{\varepsilon k\leq d_k}\frac{d_k}{\varepsilon k}.
]

(^*) In a less precise form this inequality was noted in the note ((^{16})).

({}^{**}) In some details our computations are close to estimates of the (\varepsilon)-entropy of concrete compacta in the works of V. D. Erokhin ((^{15})).

This is the fundamental inequality. From it it follows that:

[
H(K,S;\varepsilon)\geqslant \sum_{\varepsilon k<d_k}\log \frac{d_k}{\varepsilon k}
=\int_0^{1/\varepsilon}\log \frac{1}{\varepsilon t}\,dl(t)
=\int_0^{1/\varepsilon}\frac{l(t)}{t}\,dt.
]

The theorem is proved.

Let us note that the inequalities of Theorem 1 are sharp in the sense that one can indicate compact sets for which the right- or left-hand inequality turns into an asymptotic equality. We also point out that, in fact, Theorem 2 of the note (10) has been proved above.

From the last inequality it follows, in particular, that

[
H(K,S;\varepsilon)\geqslant
\int_{1/\varepsilon}^{1/\varepsilon}\frac{l(t)}{t}\,dt
\geqslant l!\left(\frac{1}{e\varepsilon}\right).
]

Using the fact that if (\sum \xi_n<\infty) and (\xi_n\downarrow 0), then (n\xi_n\to 0), from Theorem 1 and Lemma 4 we obtain:

Lemma 5. Let (\alpha) be the convergence exponent of the sequence (d_n); then (\rho(K,S)\leqslant \alpha). If (\rho(K,S)=\rho<1), then (\alpha\leqslant \rho/(1-\rho)); in particular, if (\rho(K,S)<1/4), then (\sum n^2d_n<\infty).

3. Lemma 6* (Auerbach, (11), p. 205). In every (n)-dimensional Banach space (L_n) there exists an “orthonormalized basis,” i.e. systems of elements ({g_i,\ i=1,\ldots,n}) and functionals ({g_i',\ i=1,\ldots,n}) such that (|g_i|=|g_i'|=1,\ g_i'(g_j)=\delta_{ij}) for (1\leqslant i,j\leqslant n).

If (L_n) is an (n)-dimensional subspace in a Banach space (B), and ({g_i,g_i'}) is an Auerbach basis in (L_n), then, extending (g_i') to all of (B) with preservation of the norm, we obtain a linear operator

[
P_n,\qquad P_nx=\sum_1^n g_i'(x)g_i,
]

projecting onto (L_n), with norm (|P_n|\leqslant n). The usual arguments (cf. (13), p. 193) show that

[
|x-P_nx|\leqslant (1+n)r(x,L_n)
]

for every (x\in B).

Definition 4 (see (3)). A compact set (K) is called nuclear with respect to (S) if there exists a system ({f_i,\ i=1,2,\ldots}) of linear functionals on (L(K)) and a system ({h_i,\ i=1,2,\ldots}) of vectors in (B) such that

[
\sum_1^\infty |f_i|_K\,|h_i|_S<\infty
]

and for every (x\in K) in (B) the representation

[
x=\sum_1^\infty f_i(x)h_i
]

holds. A linear topological space (E) is called nuclear if for every neighborhood (U) of zero in (E) there is a neighborhood (V) nuclear with respect to it.

The operators (P_n) constructed above make it possible to prove:

Lemma 7. If for a compact set (K) the series (\sum n^2d_n(K,S)) converges, in particular, if (\rho(K,S)<1/4), then (K) is nuclear with respect to (S).

Definition 5 (cf. (1), § 2). To every linear topological space (E) we assign a class (\Psi(E)) of functions defined for (\varepsilon>0) by the following condition: (\psi(\varepsilon)\in\Psi(E)) if

[
\forall(\delta>0)\ \forall U\ \exists V
\left{
\lim_{\varepsilon\to 0}\frac{N(V,U;\delta\varepsilon)}{\psi(\varepsilon)}=0
\right}^{**}.
]

The class (\Psi(E)) has the properties of Kolmogorov linear dimension (see (1), § 1) and, as a theorem of Bessaga and Pełczyński shows (see (14), Theorem 1), in the case of a metrizable space (\Psi(E)) coincides with the class (\Phi(E)) of functions defining approximative

* For its proof see (12).

** (\forall(\delta>0)) means “for every (\delta>0),” etc.; (\exists V) means “there exists such a neighborhood (V) of zero that …”.

dimension in the sense of Kolmogorov (see (¹), § 2). From Lemmas 7 and 2 (cf. (⁷), where Theorem 2 is proved for some metrizable spaces) it follows:

Theorem 2 (criterion of nuclearity). For the nuclearity of a linear topological space (E) it is necessary that (\exp \varepsilon^{-\lambda} \in \Psi'(E)) for all (\lambda>0), and sufficient that (\exp \varepsilon^{-\lambda_0}\in \Psi'(E)) for some (\lambda_0>0).

Another, as follows from all that has been said above, equivalent formulation of the criterion is possible:

Theorem 2a. For the nuclearity of the space (E) it is necessary that

[
\forall(\lambda>0)\ \forall U\,\exists V\,{n^\lambda d_n(V,U)\to 0},
]

and sufficient that

[
\exists(\lambda_0>0)\ \forall U\,\exists V\,{n^{\lambda_0}d_n(V,U)\to 0}.
]

In conclusion I express my gratitude to M. A. Krasnosel’skii, who pointed out Auerbach’s lemma to me, and to V. Erokhin and V. Tikhomirov for useful discussions and remarks.

Moscow State University
named after M. V. Lomonosov

Received
20 IV 1960

CITED LITERATURE

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