BANACH SPACES OF POLYNOMIALS AS âLARGEâ SUBSPACES OF
BANACH SPACES OF POLYNOMIALS AS “LARGE” SUBSPACES OF ℓ∞ -SPACES
arXiv:1402.2303v2 [math.FA] 21 Apr 2014
ALEXANDER BRUDNYI Abstract. In this note we study Banach spaces of traces of real polynomials on Rn to compact subsets equipped with supremum norms from the point of view of Geometric Functional Analysis.
1. Main Results Recall that the Banach-Mazur distance between two k-dimensional real Banach spaces E, F is defined as dBM (E, F ) := inf{kuk · ku−1 k},
where the infimum is taken over all isomorphisms u : E → F . We say that E and F are equivalent if they are isometrically isomorphic (i.e., dBM (E, F ) = 1). Then ln dBM determines a metric on the set Bk of equivalence classes of isometrically isomorphic kdimensional Banach spaces (called the Banach-Mazur compactum). It is known that Bk is compact of dBM -“diameter” ∼ k, see [G]. Let C(K) be the Banach space of real continuous functions on a compact Hausdorff space K equipped with the supremum norm. Let F ⊂ C(K) be a filtered subalgebra with filtration {0} ⊂ F0 ⊆ F1 ⊆ · · · ⊆ Fd ⊆ · · · ⊆ F (that is, F = ∪d∈Z+ Fi and Fi · Fj ⊂ Fi+j for all i, j ∈ Z+ ) such that nd := dim Fd < ∞ for all d. In what follows we assume that F0 contains constant functions on K. Our main result is Theorem 1.1. Suppose there are c ∈ R and {pd }d∈N ⊂ N such that ln nd·pd ≤c pd
(1.1)
for all
d ∈ N.
Then there exist linear injective maps id : Fd ֒→ ℓ∞ nd·p such that d
c
dBM (Fd , id (Fd )) ≤ e ,
d ∈ N.
As a corollary we obtain: Corollary 1.2. Suppose {nd }d∈N grows at most polynomially in d, that is, (1.2)
∃ k, cˆ ∈ R+ such that ∀ d
nd ≤ cˆdk .
Then for each natural number s ≥ 3 there exist linear injective maps id,s : Fd ֒→ ℓ∞ Nd,s , k j k where Nd,s := cˆdk · sk · ⌊ln(ˆ cdk )⌋ + 1 , such that 1 s dBM (Fd , id,s (Fd )) ≤ esk ,
k ∈ N.
2010 Mathematics Subject Classification. Primary 46B20, Secondary 46E15. Key words and phrases. Banach-Mazur compactum, filtered algebra, entropy. Research supported in part by NSERC. 1
2
ALEXANDER BRUDNYI
Let Fcˆ,k be the family of all possible filtered algebras F on compact Hausdorff spaces K satisfying condition (1.2). By Bcˆ,k,¯nd ⊂ Bn¯ d we denote the closure in Bn¯ d of the set formed by all subspaces Fd of algebras F ∈ Fcˆ,k having a fixed dimension n ¯ d ∈ N. Corollary 1.2 allows to estimate the metric entropy of Bcˆ,k,¯nd . Recall that for a compact subset S ⊂ Bn¯ d its ε-entropy (ε > 0) is defined as H(S, ε) := ln N (S, dBM , 1 + ε), where N (S, dBM , 1 + ε) is the smallest number of open dBM -“balls” of radius 1 + ε that cover S. Corollary 1.3. For k ≥ 1 there exists a numerical constant C such that for each ε ∈ (0, 12 ] k k+1 1 1 · ln . H Bcˆ,k,¯nd , ε ≤ (Ck · ln(k + 1))k · (ˆ cdk )2 · (ln(ˆ cdk ) + 1)k+1 · ε ε 2. Basic Example: Banach Spaces of Polynomials Let Pdn be the space of real polynomials on Rn of degree at most d. For a compact subset K ⊂ Rn by Pdn |K we denote the trace space of restrictions of polynomials in Pdn to K equipped with the supremum norm. Applying Corollary 1.2 to algebra P n |K := ∪d≥0 Pdn |K we obtain: (A) There exist linear injective maps id,K : Pdn |K ֒→ ℓ∞ Nd,n , where 2n (2.1) Nd,n := e · (n + 2)2n · dn · (2n + 1 + ⌊n ln d⌋)n , such that (2.2)
dBM (Pdn |K , id,K (Pdn |K )) ≤ e · (n + 2)2
Indeed, ed,n := dim P n |K ≤ (2.3) N d
d+n n
c1 (n) · N m ) ≤ 3. 1 2 e , then due to property (A) for each In turn, if dˆ ∈ N is such that Nd,n ˆ ≤ c1 (n) · Nd,n n ′ n e P -determining compact set K ⊂ R there exists a N ˆ -dimensional subspace F ˆ ′ ⊂ F d,n
such that (2.6)
d,n,K
n dBM Fd,n,K ′ , P ˆ |K ′ < 9. ˆ d
Further, the dual space (Vdn (K))∗ of Vdn (K) is the quotient space of ℓ1Nd,n . In particular, ed,n · 1 + ln N ed,n n extreme points, the closed ball of (Vdn (K))∗ contains at most c(n) · N see (2.4). Thus the balls of (Vdn (K))∗ and Vdn (K) are “quite different” as convex bodies. This is also expressed in the following property (similar to the celebrated John ellipsoid
BANACH SPACES OF POLYNOMIALS
3
theorem but with an extra logarithmic factor) which is a consequence of property (A) and [DMTJ, Prop. 1]: (C) There is a constant c2 (n) (depending on n only) such that for all P n -determining compact sets K1 , K2 ⊂ Rn ed,n · (1 + ln N ed,n )) 12 . (2.7) dBM Pdn |K1 , (Pdn |K2 )∗ ≤ c2 (n) · (N A stronger inequality is valid if we replace (Pdn |K2 )∗ above by ℓ1e
Nd,n
, see [DMTJ, Th. 2].
Remark 2.1. Property (C) has the following geometric interpretation. By definition, ed,n -dimensional real Banach space generated by evaluation functionals (Pdn |K2 )∗ is a N δx at points x ∈ K2 with the closed unit ball being the balanced convex hull of the set {δx }x∈K2 . Thus K2 admits a natural isometric embedding into the unit sphere of (Pdn |K2 )∗ . Moreover, the Banach space of linear maps (Pdn |K2 )∗ → Pdn |K1 equipped with the operator norm is isometrically isomorphic to the Banach space of real polynomial maps p : Rn → Pdn |K1 of degree at most d (i.e., f ∗ ◦ p ∈ Pdn for all f ∗ ∈ (Pdn |K1 )∗ ) with norm kpk := supx∈K2 kp(x)kPdn |K1 . Thus property (C) is equivalent to the following one:
(C′ ) There exists a polynomial map p : Rn → Pdn |K1 of degree at most d such that the balanced convex hull of p(K2 ) contains the closed unit ball of Pdn |K1 and is contained in ed,n · (1 + ln N ed,n )) 12 of this space (both centered at 0). the closed ball of radius c2 (n) · (N
Our next property, a consequence of Corollary 1.3 and (2.3), estimates the metric ened,n -dimensional spaces P n |K ed,n ⊂ B e formed by all N tropy of the closure of the set P d Nd,n with P n -determining compact subsets K ⊂ Rn . (D) There exists a numerical constant c > 0 such that for each ε ∈ (0, 12 ], n n+1 1 1 2 n 2n n+1 e (2.8) H cl(Pd,n ), ε ≤ (cn · ln(n + 1)) · d · (1 + ln d) · · ln . ε ε
ed,n with sufficiently large d and n is much Remark 2.2. The above estimate shows that P less massive than BNe . Indeed, as follows from [Br, Th. 2] d,n
Ned,n −1 2 1 H(BNed,n , ε) ∼ ε
as
ε → 0+
(here the equivalence depends on d and n as well). On the other hand, [P, Th. 1.1] implies that for any ε > 0, ln H(BNe , ε) ln H(BNe , ε) d,n d,n ≤ lim sup < ∞. 0 < lim inf e e e Nd,n Nd,n Nd,n →∞ ed,n →∞ N It might be of interest to find sharp asymptotics of H cl(Ped,n ), ε as ε → 0+ and d → ∞, and to compute (up to a constant depending on n) dBM -“diameter” of Ped,n .
Similar results are valid for K being a compact subset of a real algebraic variety X ⊂ Rn of dimension m < n such that if a polynomial vanishes on K, then it vanishes on X as well. In this case there are positive constants cX , c˜X depending on X only such that c˜X dm ≤ dim Pdn |K ≤ cX dm . For instance, Corollary 1.2 with c = cX , k := m and s := (m + 2)2 implies that Pdn |K is linearly embedded into ℓ∞ Nd,X , where Nd,X := m m 2m m cX d · (m + 2) · (⌊ln(cX d )⌋ + 1) , with distortion < 2.903. We leave the details to the reader.
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ALEXANDER BRUDNYI
3. Proofs Proof of Theorem 1.1. Since dim Fi = ni , i ∈ N, and evaluations δz at points z ∈ K determine bounded linear functionals on Fi , the Hahn-Banach theorem implies easily that span {δz }z∈K = Fi∗ . Moreover, kδz kFi∗ = 1 for all z ∈ K and the closed unit ball of Fi∗ is the balanced convex hull of the set {δz }z∈K . Let {f1i , . . . , fni i } ⊂ Fi be an Auerbach basis with the dual basis {δz1i , . . . , δzni i } ⊂ Fi∗ , that is, fki (δzli ) := fki (zli ) = δkl (the Kroneckerdelta) and kfki kK = 1 for all k. (Its construction is similar to that of the fundamental Lagrange interpolation polynomials for Fi = Pin |K , see, e.g., [BY, Prop. 2.2].) Now, we use a “method of E. Landau” (see, e.g., [PS, 3, § 2]). PCh. ni By the definition, for each g ∈ Fi we have g(z) = k=1 fki (z)g(zki ), z ∈ K. Hence, kgkK ≤ ni kgk{z1i ,...,zni i } . Applying the latter inequality to g = f pd , f ∈ Fd , containing in Fi , i := d · pd , and using condition (1.1) we get for Ad := {z1i , . . . , zni i } ⊂ K 1
1
1
kf kK = (kgkK ) pd ≤ (nd·pd ) pd · (kgkAd ) pd ≤ ec · kf kAd . Thus, restriction Fd 7→ Fd |Ad determines the required map id : Fd ֒→ ℓ∞ nd·p . d
Proof of Corollary 1.2. We set pd := s · (⌊ln(ˆ cdk )⌋ + 1), d ∈ N. Then the condition of the corollary implies ln nd·pd ln(ˆ cdk ) + k ln pd 1 k ln s ≤ ≤ + =: c. pd pd s s Thus the result follows from Theorem 1.1. Proof of Corollary 1.3. We make use of [P, Lm. 1.2] adapted to our setting: Lemma 3.1. Let Sn¯ d ⊂ Bn¯ d be the subset formed by all n ¯ d -dimensional subspaces of ℓ∞ Nd,s .
Consider 0 < ξ < n¯1d and let R = Nd,s ·¯nd most 1 + 2ξ .
1+ξ¯ nd 1−ξ¯ nd .
Then Sn¯ d admits an R-net TR of cardinality at
1 √ Now given ε ∈ (0, 12 ] we choose s = ⌊sε ⌋ with sε satisfying (eskε ) sε = 4 1 + ε and √ ξ such that R = Rε =√ 4 1 + ε. Then according to Corollary 1.2 and Lemma 3.1, distBM TRε , Bcˆ,k,¯nd < 1 + ε. For each p ∈ TRε we choose qp ∈ Bcˆ,k,¯nd such that √ dBM (p, qp ) < 1 + ε. Then the multiplicative triangle inequality for dBM implies that open dBM -“balls” of radius 1 + ε centered at points qp , p ∈ TRε , cover Bcˆ,k,¯nd . Hence, 2 Nd,s ·¯nd (3.9) N (Bcˆ,k,¯nd , dBM , 1 + ε) ≤ card TRε ≤ 1 + . ξ k−1
1
Next, the function ϕ(x) = ln(exk ) x decreases for x ∈ [e k , ∞) and limx→∞ ϕ(x) = 0. k−1 Its inverse ϕ−1 on this interval has domain (0, e− k ], increases and is easily seen (using that ϕ ◦ ϕ−1 = id) to satisfy k−1 3k 3k −1 · ln , x ∈ (0, e− k ]. ϕ (x) ≤ x x Since 41 ln(1 + ε) < e− implies that (3.10)
k−1 k
for ε ∈ (0, 12 ], the required sε exists and the previous inequality 12k sε ≤ · ln ln(1 + ε)
12k ln(1 + ε)
.
BANACH SPACES OF POLYNOMIALS
5
Further, we have
√ √ √ 1 n ¯ d (1 + Rε ) n ¯ d ( 4 1 + ε + 1) n ¯ d ( 4 1 + ε + 1)2 · ( 1 + ε + 1) = = √ . = 4 ξ Rε − 1 ε 1+ε−1 From (3.9), (3.10), (3.11) invoking the definition of Nd,s we obtain k k 21¯ 12k 12k nd k k ln . cd ) + 1 ln ln N (Bcˆ,k,¯nd , dBM , 1 + ε) ≤ n ¯ d cˆd ln(ˆ ε ln(1 + ε) ln(1 + ε) (3.11)
Using that n ¯ d ≤ cˆdk and the inequality estimate.
2 3
· ε ≤ ln(1 + ε), ε ∈ (0, 12 ], we get the required
References [B] J. Bourgain, Subspaces of L∞ N , arithmetical diameter and Sidon sets. Probability in Banach Spaces V. Proceed. Medford 1984. Lecture Notes in Math. 1153 (1985), 96–127. [Br] E. M. Bronstein, ε-entropy of affine-equivalent convex bodies and Minkowski’s compactum. (Russian) Optimizatsiya 39 No. 22 (1978), 5–11, 155. [BY] A. Brudnyi and Y. Yomdin, Norming sets and related Remez-type inequalities. arXiv:1312.6050. [G] E. D. Gluskin, The diameter of the Minkowski compactum is roughly equal to n. Funktsional. Anal. i Prilozhen. 15 (1) (1981), 72–73. [DMTJ] W. J. Davis, V. D. Milman and N. Tomczak-Jaegermann, The distance between certain ndimensional Banach spaces. Israel J. Math. 39 (1981), 1–15. [P] G. Pisier, On the metric entropy of the Banach-Mazur compactum. arXiv:1306.5325v6. [PS] G. Polya and G. Szeg¨ o, Problems and theorems in Analysis II. Springer, 1998. Department of Mathematics and Statistics University of Calgary Calgary, Alberta T2N 1N4 E-mail address: [email protected]
arXiv:1402.2303v2 [math.FA] 21 Apr 2014
ALEXANDER BRUDNYI Abstract. In this note we study Banach spaces of traces of real polynomials on Rn to compact subsets equipped with supremum norms from the point of view of Geometric Functional Analysis.
1. Main Results Recall that the Banach-Mazur distance between two k-dimensional real Banach spaces E, F is defined as dBM (E, F ) := inf{kuk · ku−1 k},
where the infimum is taken over all isomorphisms u : E → F . We say that E and F are equivalent if they are isometrically isomorphic (i.e., dBM (E, F ) = 1). Then ln dBM determines a metric on the set Bk of equivalence classes of isometrically isomorphic kdimensional Banach spaces (called the Banach-Mazur compactum). It is known that Bk is compact of dBM -“diameter” ∼ k, see [G]. Let C(K) be the Banach space of real continuous functions on a compact Hausdorff space K equipped with the supremum norm. Let F ⊂ C(K) be a filtered subalgebra with filtration {0} ⊂ F0 ⊆ F1 ⊆ · · · ⊆ Fd ⊆ · · · ⊆ F (that is, F = ∪d∈Z+ Fi and Fi · Fj ⊂ Fi+j for all i, j ∈ Z+ ) such that nd := dim Fd < ∞ for all d. In what follows we assume that F0 contains constant functions on K. Our main result is Theorem 1.1. Suppose there are c ∈ R and {pd }d∈N ⊂ N such that ln nd·pd ≤c pd
(1.1)
for all
d ∈ N.
Then there exist linear injective maps id : Fd ֒→ ℓ∞ nd·p such that d
c
dBM (Fd , id (Fd )) ≤ e ,
d ∈ N.
As a corollary we obtain: Corollary 1.2. Suppose {nd }d∈N grows at most polynomially in d, that is, (1.2)
∃ k, cˆ ∈ R+ such that ∀ d
nd ≤ cˆdk .
Then for each natural number s ≥ 3 there exist linear injective maps id,s : Fd ֒→ ℓ∞ Nd,s , k j k where Nd,s := cˆdk · sk · ⌊ln(ˆ cdk )⌋ + 1 , such that 1 s dBM (Fd , id,s (Fd )) ≤ esk ,
k ∈ N.
2010 Mathematics Subject Classification. Primary 46B20, Secondary 46E15. Key words and phrases. Banach-Mazur compactum, filtered algebra, entropy. Research supported in part by NSERC. 1
2
ALEXANDER BRUDNYI
Let Fcˆ,k be the family of all possible filtered algebras F on compact Hausdorff spaces K satisfying condition (1.2). By Bcˆ,k,¯nd ⊂ Bn¯ d we denote the closure in Bn¯ d of the set formed by all subspaces Fd of algebras F ∈ Fcˆ,k having a fixed dimension n ¯ d ∈ N. Corollary 1.2 allows to estimate the metric entropy of Bcˆ,k,¯nd . Recall that for a compact subset S ⊂ Bn¯ d its ε-entropy (ε > 0) is defined as H(S, ε) := ln N (S, dBM , 1 + ε), where N (S, dBM , 1 + ε) is the smallest number of open dBM -“balls” of radius 1 + ε that cover S. Corollary 1.3. For k ≥ 1 there exists a numerical constant C such that for each ε ∈ (0, 12 ] k k+1 1 1 · ln . H Bcˆ,k,¯nd , ε ≤ (Ck · ln(k + 1))k · (ˆ cdk )2 · (ln(ˆ cdk ) + 1)k+1 · ε ε 2. Basic Example: Banach Spaces of Polynomials Let Pdn be the space of real polynomials on Rn of degree at most d. For a compact subset K ⊂ Rn by Pdn |K we denote the trace space of restrictions of polynomials in Pdn to K equipped with the supremum norm. Applying Corollary 1.2 to algebra P n |K := ∪d≥0 Pdn |K we obtain: (A) There exist linear injective maps id,K : Pdn |K ֒→ ℓ∞ Nd,n , where 2n (2.1) Nd,n := e · (n + 2)2n · dn · (2n + 1 + ⌊n ln d⌋)n , such that (2.2)
dBM (Pdn |K , id,K (Pdn |K )) ≤ e · (n + 2)2
Indeed, ed,n := dim P n |K ≤ (2.3) N d
d+n n
c1 (n) · N m ) ≤ 3. 1 2 e , then due to property (A) for each In turn, if dˆ ∈ N is such that Nd,n ˆ ≤ c1 (n) · Nd,n n ′ n e P -determining compact set K ⊂ R there exists a N ˆ -dimensional subspace F ˆ ′ ⊂ F d,n
such that (2.6)
d,n,K
n dBM Fd,n,K ′ , P ˆ |K ′ < 9. ˆ d
Further, the dual space (Vdn (K))∗ of Vdn (K) is the quotient space of ℓ1Nd,n . In particular, ed,n · 1 + ln N ed,n n extreme points, the closed ball of (Vdn (K))∗ contains at most c(n) · N see (2.4). Thus the balls of (Vdn (K))∗ and Vdn (K) are “quite different” as convex bodies. This is also expressed in the following property (similar to the celebrated John ellipsoid
BANACH SPACES OF POLYNOMIALS
3
theorem but with an extra logarithmic factor) which is a consequence of property (A) and [DMTJ, Prop. 1]: (C) There is a constant c2 (n) (depending on n only) such that for all P n -determining compact sets K1 , K2 ⊂ Rn ed,n · (1 + ln N ed,n )) 12 . (2.7) dBM Pdn |K1 , (Pdn |K2 )∗ ≤ c2 (n) · (N A stronger inequality is valid if we replace (Pdn |K2 )∗ above by ℓ1e
Nd,n
, see [DMTJ, Th. 2].
Remark 2.1. Property (C) has the following geometric interpretation. By definition, ed,n -dimensional real Banach space generated by evaluation functionals (Pdn |K2 )∗ is a N δx at points x ∈ K2 with the closed unit ball being the balanced convex hull of the set {δx }x∈K2 . Thus K2 admits a natural isometric embedding into the unit sphere of (Pdn |K2 )∗ . Moreover, the Banach space of linear maps (Pdn |K2 )∗ → Pdn |K1 equipped with the operator norm is isometrically isomorphic to the Banach space of real polynomial maps p : Rn → Pdn |K1 of degree at most d (i.e., f ∗ ◦ p ∈ Pdn for all f ∗ ∈ (Pdn |K1 )∗ ) with norm kpk := supx∈K2 kp(x)kPdn |K1 . Thus property (C) is equivalent to the following one:
(C′ ) There exists a polynomial map p : Rn → Pdn |K1 of degree at most d such that the balanced convex hull of p(K2 ) contains the closed unit ball of Pdn |K1 and is contained in ed,n · (1 + ln N ed,n )) 12 of this space (both centered at 0). the closed ball of radius c2 (n) · (N
Our next property, a consequence of Corollary 1.3 and (2.3), estimates the metric ened,n -dimensional spaces P n |K ed,n ⊂ B e formed by all N tropy of the closure of the set P d Nd,n with P n -determining compact subsets K ⊂ Rn . (D) There exists a numerical constant c > 0 such that for each ε ∈ (0, 12 ], n n+1 1 1 2 n 2n n+1 e (2.8) H cl(Pd,n ), ε ≤ (cn · ln(n + 1)) · d · (1 + ln d) · · ln . ε ε
ed,n with sufficiently large d and n is much Remark 2.2. The above estimate shows that P less massive than BNe . Indeed, as follows from [Br, Th. 2] d,n
Ned,n −1 2 1 H(BNed,n , ε) ∼ ε
as
ε → 0+
(here the equivalence depends on d and n as well). On the other hand, [P, Th. 1.1] implies that for any ε > 0, ln H(BNe , ε) ln H(BNe , ε) d,n d,n ≤ lim sup < ∞. 0 < lim inf e e e Nd,n Nd,n Nd,n →∞ ed,n →∞ N It might be of interest to find sharp asymptotics of H cl(Ped,n ), ε as ε → 0+ and d → ∞, and to compute (up to a constant depending on n) dBM -“diameter” of Ped,n .
Similar results are valid for K being a compact subset of a real algebraic variety X ⊂ Rn of dimension m < n such that if a polynomial vanishes on K, then it vanishes on X as well. In this case there are positive constants cX , c˜X depending on X only such that c˜X dm ≤ dim Pdn |K ≤ cX dm . For instance, Corollary 1.2 with c = cX , k := m and s := (m + 2)2 implies that Pdn |K is linearly embedded into ℓ∞ Nd,X , where Nd,X := m m 2m m cX d · (m + 2) · (⌊ln(cX d )⌋ + 1) , with distortion < 2.903. We leave the details to the reader.
4
ALEXANDER BRUDNYI
3. Proofs Proof of Theorem 1.1. Since dim Fi = ni , i ∈ N, and evaluations δz at points z ∈ K determine bounded linear functionals on Fi , the Hahn-Banach theorem implies easily that span {δz }z∈K = Fi∗ . Moreover, kδz kFi∗ = 1 for all z ∈ K and the closed unit ball of Fi∗ is the balanced convex hull of the set {δz }z∈K . Let {f1i , . . . , fni i } ⊂ Fi be an Auerbach basis with the dual basis {δz1i , . . . , δzni i } ⊂ Fi∗ , that is, fki (δzli ) := fki (zli ) = δkl (the Kroneckerdelta) and kfki kK = 1 for all k. (Its construction is similar to that of the fundamental Lagrange interpolation polynomials for Fi = Pin |K , see, e.g., [BY, Prop. 2.2].) Now, we use a “method of E. Landau” (see, e.g., [PS, 3, § 2]). PCh. ni By the definition, for each g ∈ Fi we have g(z) = k=1 fki (z)g(zki ), z ∈ K. Hence, kgkK ≤ ni kgk{z1i ,...,zni i } . Applying the latter inequality to g = f pd , f ∈ Fd , containing in Fi , i := d · pd , and using condition (1.1) we get for Ad := {z1i , . . . , zni i } ⊂ K 1
1
1
kf kK = (kgkK ) pd ≤ (nd·pd ) pd · (kgkAd ) pd ≤ ec · kf kAd . Thus, restriction Fd 7→ Fd |Ad determines the required map id : Fd ֒→ ℓ∞ nd·p . d
Proof of Corollary 1.2. We set pd := s · (⌊ln(ˆ cdk )⌋ + 1), d ∈ N. Then the condition of the corollary implies ln nd·pd ln(ˆ cdk ) + k ln pd 1 k ln s ≤ ≤ + =: c. pd pd s s Thus the result follows from Theorem 1.1. Proof of Corollary 1.3. We make use of [P, Lm. 1.2] adapted to our setting: Lemma 3.1. Let Sn¯ d ⊂ Bn¯ d be the subset formed by all n ¯ d -dimensional subspaces of ℓ∞ Nd,s .
Consider 0 < ξ < n¯1d and let R = Nd,s ·¯nd most 1 + 2ξ .
1+ξ¯ nd 1−ξ¯ nd .
Then Sn¯ d admits an R-net TR of cardinality at
1 √ Now given ε ∈ (0, 12 ] we choose s = ⌊sε ⌋ with sε satisfying (eskε ) sε = 4 1 + ε and √ ξ such that R = Rε =√ 4 1 + ε. Then according to Corollary 1.2 and Lemma 3.1, distBM TRε , Bcˆ,k,¯nd < 1 + ε. For each p ∈ TRε we choose qp ∈ Bcˆ,k,¯nd such that √ dBM (p, qp ) < 1 + ε. Then the multiplicative triangle inequality for dBM implies that open dBM -“balls” of radius 1 + ε centered at points qp , p ∈ TRε , cover Bcˆ,k,¯nd . Hence, 2 Nd,s ·¯nd (3.9) N (Bcˆ,k,¯nd , dBM , 1 + ε) ≤ card TRε ≤ 1 + . ξ k−1
1
Next, the function ϕ(x) = ln(exk ) x decreases for x ∈ [e k , ∞) and limx→∞ ϕ(x) = 0. k−1 Its inverse ϕ−1 on this interval has domain (0, e− k ], increases and is easily seen (using that ϕ ◦ ϕ−1 = id) to satisfy k−1 3k 3k −1 · ln , x ∈ (0, e− k ]. ϕ (x) ≤ x x Since 41 ln(1 + ε) < e− implies that (3.10)
k−1 k
for ε ∈ (0, 12 ], the required sε exists and the previous inequality 12k sε ≤ · ln ln(1 + ε)
12k ln(1 + ε)
.
BANACH SPACES OF POLYNOMIALS
5
Further, we have
√ √ √ 1 n ¯ d (1 + Rε ) n ¯ d ( 4 1 + ε + 1) n ¯ d ( 4 1 + ε + 1)2 · ( 1 + ε + 1) = = √ . = 4 ξ Rε − 1 ε 1+ε−1 From (3.9), (3.10), (3.11) invoking the definition of Nd,s we obtain k k 21¯ 12k 12k nd k k ln . cd ) + 1 ln ln N (Bcˆ,k,¯nd , dBM , 1 + ε) ≤ n ¯ d cˆd ln(ˆ ε ln(1 + ε) ln(1 + ε) (3.11)
Using that n ¯ d ≤ cˆdk and the inequality estimate.
2 3
· ε ≤ ln(1 + ε), ε ∈ (0, 12 ], we get the required
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