on perfectly homogeneous bases in quasi-banach...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2009, Article ID 865371, 7 pagesdoi:10.1155/2009/865371

Research ArticleOn Perfectly Homogeneous Bases inQuasi-Banach Spaces

F. Albiac and C. Leranoz

Departamento de Matematicas, Universidad Publica de Navarra, 31006 Pamplona, Spain

Correspondence should be addressed to F. Albiac, [email protected]

Received 22 April 2009; Accepted 3 June 2009

Recommended by Simeon Reich

For 0 < p < ∞ the unit vector basis of �p has the property of perfect homogeneity: it is equivalentto all its normalized block basic sequences, that is, perfectly homogeneous bases are a special caseof symmetric bases. For Banach spaces, a classical result of Zippin (1966) proved that perfectlyhomogeneous bases are equivalent to either the canonical c0-basis or the canonical �p-basis forsome 1 ≤ p < ∞. In this note, we show that (a relaxed form of) perfect homogeneity characterizesthe unit vector bases of �p for 0 < p < 1 as well amongst bases in nonlocally convex quasi-Banachspaces.

Copyright q 2009 F. Albiac and C. Leranoz. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction and Background

Let us first review the relevant elementary concepts and definitions. Further details can befound in the books [1, 2] and the paper [3]. A (real) quasi-normed space X is a locallybounded topological vector space. This is equivalent to saying that the topology on X isinduced by a quasi-norm, that is, a map ‖ · ‖ : X → [0,∞) satisfying

(i) ‖x‖ = 0 if and only if x = 0;

(ii) ‖αx‖ = |α|‖x‖ if α ∈ R, x ∈ X;

(iii) there is a constant κ ≥ 1 so that for any x1 and x2 ∈ X we have

‖x1 + x2‖ ≤ κ(‖x1‖ + ‖x2‖). (1.1)

The best constant κ in inequality (1.1) is called the modulus of concavity of the quasi-norm. Ifκ = 1, the quasi-norm is a norm. A quasi-norm on X is p-subadditive if

‖x1 + x2‖p ≤ ‖x1‖p + ‖x2‖p, x1, x2 ∈ X. (1.2)

2 Abstract and Applied Analysis

A theorem by Aoki [4] and Rolewicz [5] asserts that every quasi-norm has an equivalent p-subadditive quasi-norm, where 0 < p ≤ 1 is given by κ = 21/p−1. A p-subadditive quasi-norm‖ · ‖ induces an invariant metric on X by the formula d(x, y) = ‖x − y‖p. The space X is calledquasi-Banach space if X is complete for this metric. A quasi-Banach space is isomorphic to aBanach space if and only if it is locally convex.

A basis (xn)∞n=1 of a quasi-Banach space X is symmetric if (xn)

∞n=1 is equivalent to

(xπ(n))∞n=1 for any permutation π of N. Symmetric bases are unconditional and so there exists

a nonnegative constant K such that for all x =∑∞

n=1anxn the inequality

∥∥∥∥∥

∞∑

n=1

θnanxn

∥∥∥∥∥≤ K

∥∥∥∥∥

∞∑

n=1

anxn

∥∥∥∥∥

(1.3)

holds for any bounded sequence (θn)∞n=1 ∈ B�∞ . The least such constant K is called the

unconditional constant of (xn)∞n=1.

For instance, the canonical basis of the spaces �p for 0 < p < ∞ is symmetric and 1-unconditional. What is more, it is the only symmetric basis of �p up to equivalence, that is,whenever (xn)

∞n=1 is another normalized symmetric basis of �p, there is a constant C such that

1C

( ∞∑

n=1

|an|p)1/p

≤∥∥∥∥∥

∞∑

n=1

anxn

∥∥∥∥∥≤ C

( ∞∑

n=1

|an|p)1/p

, (1.4)

for any finitely nonzero sequence of scalars (an)∞n=1 [6, 7].

The spaces �p for 0 < p < 1 share the property of uniqueness of symmetric basis withall natural quasi-Banach spaces whose Banach envelope (i.e., the smallest containing Banachspace) is isomorphic to �1, as was recently proved in [8]. For other results on uniqueness ofunconditional or symmetric basis in nonlocally convex quasi-Banach spaces the reader canconsult the papers [9, 10].

This article illustrates howZippin’s techniques can also be used to characterize the unitvector bases of �p for 0 < p < 1 as the only, up to equivalence, perfectly homogeneous bases innonlocally convex quasi-Banach spaces. We use standard Banach space theory terminologyand notation throughout, as may be found in [11, 12].

2. Perfectly Homogeneous Bases in Quasi-Banach Spaces

Let (xi)∞i=1 be a basis for a quasi-Banach space X. A block basic sequence (un)

∞n=1 of (xi)

∞i=1,

un =pn∑

pn−1+1

aixi, (2.1)

is said to be a constant coefficient block basic sequence if for each n there is a constant cn so thatai = cn or ai = 0 for pn−1 + 1 ≤ i ≤ pn.

Definition 2.1. A basis (xi)∞i=1 of a quasi-Banach spaceX is almost perfectly homogeneous if every

normalized constant coefficient block basic sequence of (xi)∞i=1 is equivalent to (xi)

∞i=1.

Abstract and Applied Analysis 3

Let us notice that using a uniform boundedness argument we obtain that, in fact, if(xi)

∞i=1 is almost perfectly homogeneous then it is uniformly equivalent to all its normalized

constant coefficient block basic sequences. That is, there is a constantM ≥ 1 such that for anynormalized constant coefficient block basic sequence (un)

∞n=1 of (xi)

∞i=1 we have

M−1∥∥∥∥∥

n∑

k=1

akxk

∥∥∥∥∥≤∥∥∥∥∥

n∑

k=1

akuk

∥∥∥∥∥≤ M

∥∥∥∥∥

n∑

k=1

akxk

∥∥∥∥∥, (2.2)

for all choices of scalars (ak)nk=1 and n ∈ N. Equation (2.2) also yields that for any increasing

sequence of integers (kj)∞j=1,

M−1

∥∥∥∥∥∥

n∑

j=1

xj

∥∥∥∥∥∥≤∥∥∥∥∥∥

n∑

j=1

xkj

∥∥∥∥∥∥≤ M

∥∥∥∥∥∥

n∑

j=1

xj

∥∥∥∥∥∥. (2.3)

This is our main result (cf. [13]).

Theorem 2.2. Let X be a nonlocally convex quasi-Banach space with normalized basis (xi)∞i=1.

Suppose that (xi)∞i=1 is almost perfectly homogeneous. Then (xi)

∞i=1 is equivalent to the canonical basis

of �q for some 0 < q < 1.

Proof. Let κ be the modulus of concavity of the quasi-norm. Since X is nonlocally convex,κ > 1. By the Aoki-Rolewicz theorem we can assume that the quasi-norm is p-subadditive for0 < p < 1 such that κ = 21/p−1. We will show that (xi)

∞i=1 is equivalent to the canonical �q-basis

for some p ≤ q < 1.By renorming, without loss of generality we can assume (xi)

∞i=1 to be 1-unconditional.

For each n put,

λ(n) =

∥∥∥∥∥

n∑

i=1

xi

∥∥∥∥∥. (2.4)

Note that

1 ≤ λ(n) ≤ n1/p, n ∈ N, (2.5)

and that, by the 1-unconditionality of the basis, the sequence (λ(n))∞n=1 is nondecreasing.We are going to construct disjoint blocks of length n of the basis (xi)

∞i=1 as follows:

v1 =n∑

i=1

xi, v2 =2n∑

i=n+1

xi, . . . , vj =jn∑

i=(j−1)n+1xi, . . . . (2.6)

Equation (2.3) says that

M−1λ(n) ≤ ‖vj‖ ≤ Mλ(n), j ∈ N, (2.7)


and so by the 1-unconditionality of (xi)∞i=1,

1Mλ(n)

∥∥∥∥∥∥

m∑

j=1

vj

∥∥∥∥∥∥≤∥∥∥∥∥∥

m∑

j=1

‖vj‖−1vj

∥∥∥∥∥∥≤ M

λ(n)

∥∥∥∥∥∥

m∑

j=1

vj

∥∥∥∥∥∥, m ∈ N. (2.8)

On the other hand, by (2.2) we know that

λ(m)M

≤∥∥∥∥∥∥

m∑

j=1

‖vj‖−1vj

∥∥∥∥∥∥≤ Mλ(m), m ∈ N. (2.9)

If we put these last two inequalities together we obtain

1M2

λ(m)λ(n) ≤ λ(mn) ≤ M2λ(m)λ(n), m, n ∈ N. (2.10)

Substituting in (2.10) integers of the form m = 2k and n = 2j give

1M2

λ(2k)λ(2j)≤ λ

(2j+k

)≤ M2λ

(2k)λ(2j), k, j ∈ N. (2.11)

For k = 0, 1, 2, . . . , let h(k) = log2λ(2k). From (2.11) it follows that

∣∣h(j) − h(k) − h

(j + k

)∣∣ ≤ 2log2M. (2.12)

We need the following well-known lemma from real analysis.

Lemma 2.3. Suppose that (sn)∞n=1 is a sequence of real numbers such that

|sm+n − sm − sn| ≤ 1 (2.13)

for all m,n ∈ N. Then there is a constant c so that

|sn − cn| ≤ 1, n = 1, 2, . . . . (2.14)

Lemma 2.3 yields a constant c so that

|h(k) − ck| ≤ 2 log2M, k = 1, 2, . . . . (2.15)

In turn, using (2.5) we have

1 ≤ λ(2k)≤ 2k/p, k = 1, 2, . . . (2.16)


which implies

0 ≤ h(k) ≤ k

p, (2.17)

and so, combining with (2.15) we obtain that the range of possible values for c is

0 ≤ c ≤ 1p. (2.18)

If c = 0 then (λ(n))∞n=1 would be (uniformly) bounded and so (xi)∞i=1 would be equivalent

to the canonical basis of c0, a contradiction with the local nonconvexity of X. Otherwise, if0 < c ≤ 1/p there is q ∈ [p,∞) such that c = 1/q. This way we can rewrite (2.15) in the form

∣∣∣∣h(k) −

k

q

∣∣∣∣ ≤ 2 log2M, k ∈ N, (2.19)

or equivalently,

M−22k/q ≤ λ(2k)≤ 2k/qM2, k ∈ N. (2.20)

Now, given n ∈ N we pick the only integer k so that 2k−1 ≤ n ≤ 2k. Then,

λ(2k−1

)≤ λ(n) ≤ λ

(2k), (2.21)

and so

M−22−1/qn1/q ≤ λ(n) ≤ M221/qn1/q. (2.22)

If A is any finite subset of N, by (2.3)we have

M−1λ(|A|) ≤∥∥∥∥∥∥

∑

j∈Axj

∥∥∥∥∥∥≤ Mλ(|A|), (2.23)

hence

C−1|A|1/q ≤∥∥∥∥∥∥

∑

j∈Axj

∥∥∥∥∥∥≤ C|A|1/q, (2.24)

where C = M321/q.


To prove the equivalence of (xi)∞i=1 with the canonical basis of �q, given any n ∈ N we

let (ai)ni=1 be nonnegative scalars such that aq

i ∈ Q and∑n

i=1aq

i = 1. Each aq

i can be written inthe form a

q

i = mi/mwheremi ∈ N,m is de common denominator of the aq

i ’s, and∑n

i=1mi = m.Let A1 be interval of natural numbers [1, m1] and for j = 2, . . . , n let Ai be the interval

of natural numbers [m1 + · · ·+mi−1 +1, m1 + · · ·+mi]. The setsA1, . . . , An are disjoint and havecardinality |Ai| = mi for each i = 1, . . . , n. Consider the normalized constant coefficient blockbasic sequence defined as

ui = c−1i∑

j∈Ai

xj , 1 ≤ i ≤ n, (2.25)

where ci = ‖∑j∈Aixk‖. Equation (2.24) yields

C−1m1/qi ≤ ci ≤ Cm

1/qi , 1 ≤ i ≤ n. (2.26)

Therefore,

C−1

m1/q

∥∥∥∥∥∥

n∑

i=1

∑

j∈Ai

xj

∥∥∥∥∥∥≤∥∥∥∥∥

n∑

i=1

aiui

∥∥∥∥∥≤ C

m1/q

∥∥∥∥∥∥

n∑

i=1

∑

j∈Ai

xk

∥∥∥∥∥∥, (2.27)

that is,

C−1 λ(m)m1/q

≤∥∥∥∥∥

n∑

i=1

aiui

∥∥∥∥∥≤ C

λ(m)m1/q

. (2.28)

Thus,

1C2M

≤∥∥∥∥∥

n∑

i=1

aiui

∥∥∥∥∥≤ C2M. (2.29)

Using (2.2) again, we have

1C2M2

≤∥∥∥∥∥

n∑

i=1

aixi

∥∥∥∥∥≤ C2M2. (2.30)

We note that a simple density argument shows that (2.30) holds whenever∑n

i=1|ai|q = 1(i.e., without the assumption that |ai|q is rational), and this completes the proof that (xi)

∞i=1

is equivalent to the canonical �q-basis for some p ≤ q < ∞. Since X is not locally convex, weconclude that p ≤ q < 1.


Acknowledgment

The authors would like to acknowledge support from the Spanish Ministerio de Educaciony Ciencia Research Project Espacios Topologicos Ordenados: Resultados Analıticos y AplicacionesMultidisciplinares, reference number MTM2007-62499.

References

[1] N. J. Kalton, N. T. Peck, and J. W. Rogers, An F-Space Sampler, vol. 89 of London Mathematical SocietyLecture Note, Cambridge University Press, Cambridge, UK, 1985.

[2] S. Rolewicz, Metric Linear Spaces, vol. 20 of Mathematics and Its Applications (East European Series), D.Reidel, Dordrecht, The Netherlands, 2nd edition, 1985.

[3] N. J. Kalton, “Quasi-Banach spaces,” inHandbook of the Geometry of Banach Spaces, Vol. 2, pp. 1099–1130,North-Holland, Amsterdam, The Netherlands, 2003.

[4] T. Aoki, “Locally bounded linear topological spaces,” Proceedings of the Imperial Academy, Tokyo, vol.18, pp. 588–594, 1942.

[5] S. Rolewicz, “On a certain class of linear metric spaces,” Bulletin de L’Academie Polonaise des Sciences,vol. 5, pp. 471–473, 1957.

[6] N. J. Kalton, “Orlicz sequence spaces without local convexity,” Mathematical Proceedings of theCambridge Philosophical Society, vol. 81, no. 2, pp. 253–277, 1977.

[7] J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,” Israel Journal of Mathematics, vol. 10, pp.379–390, 1971.

[8] F. Albiac and C. Leranoz, “Uniqueness of symmetric basis in quasi-Banach spaces,” Journal ofMathematical Analysis and Applications, vol. 348, no. 1, pp. 51–54, 2008.

[9] F. Albiac and C. Leranoz, “Uniqueness of unconditional basis in Lorentz sequence spaces,” Proceedingsof the American Mathematical Society, vol. 136, no. 5, pp. 1643–1647, 2008.

[10] N. J. Kalton, C. Leranoz, and P. Wojtaszczyk, “Uniqueness of unconditional bases in quasi-Banachspaces with applications to Hardy spaces,” Israel Journal of Mathematics, vol. 72, no. 3, pp. 299–311,1990.

[11] F Albiac and N. J. Kalton, Topics in Banach Space Theory, vol. 233 of Graduate Texts in Mathematics,Springer, New York, NY, USA, 2006.

[12] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I: Sequence Spaces, Ergebnisse der Mathematikund ihrer Grenzgebiete, vol. 9, Springer, Berlin, Germany, 1977.

[13] M. Zippin, “On perfectly homogeneous bases in Banach spaces,” Israel Journal of Mathematics, vol. 4,pp. 265–272, 1966.

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