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

1. Introduction and Background

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

∥x∥=0 if and only if x=0;

∥αx∥=|α|∥x∥ if α∈ℝ,x∈X;

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

The best constant κ in inequality (1.1) is called the modulus of concavity of the quasi-norm. If κ=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.
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 called quasi-Banach space if X is complete for this metric. A quasi-Banach space is isomorphic to a Banach 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=1∞anxn the inequality
∥∑n=1∞θnanxn∥≤K∥∑n=1∞anxn∥
holds for any bounded sequence (θn)n=1∞∈Bℓ∞. The least such constant K is called the unconditional constant of (xn)n=1∞.

For instance, the canonical basis of the spaces ℓp for 0<p<∞ is symmetric and 1-unconditional. What is more, it is the only symmetric basis of ℓp up to equivalence, that is, whenever (xn)n=1∞ is another normalized symmetric basis of ℓp, there is a constant C such that
1C(∑n=1∞|an|p)1/p≤∥∑n=1∞anxn∥≤C(∑n=1∞|an|p)1/p,
for any finitely nonzero sequence of scalars (an)n=1∞ [6, 7].

The spaces ℓp for 0<p<1 share the property of uniqueness of symmetric basis with all natural quasi-Banach spaces whose Banach envelope (i.e., the smallest containing Banach space) is isomorphic to ℓ1, as was recently proved in [8]. For other results on uniqueness of unconditional or symmetric basis in nonlocally convex quasi-Banach spaces the reader can consult the papers [9, 10].

This article illustrates how Zippin’s techniques can also be used to characterize the unit vector bases of ℓp for 0<p<1 as the only, up to equivalence, perfectly homogeneous bases in nonlocally convex quasi-Banach spaces. We use standard Banach space theory terminology and 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-1+1pnaixi,
is said to be a constant coefficient block basic sequence if for each n there is a constant cn so that ai=cn or ai=0 for pn-1+1≤i≤pn.

Definition 2.1.

A basis (xi)i=1∞ of a quasi-Banach space X is almost perfectly homogeneous if every normalized constant coefficient block basic sequence of (xi)i=1∞ is equivalent to (xi)i=1∞.

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 constant M≥1 such that for any normalized constant coefficient block basic sequence (un)n=1∞ of (xi)i=1∞ we have
M-1∥∑k=1nakxk∥≤∥∑k=1nakuk∥≤M∥∑k=1nakxk∥,
for all choices of scalars (ak)k=1n and n∈ℕ. Equation (2.2) also yields that for any increasing sequence of integers (kj)j=1∞,
M-1∥∑j=1nxj∥≤∥∑j=1nxkj∥≤M∥∑j=1nxj∥.

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 ℓq 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 for 0<p<1 such that κ=21/p-1. We will show that (xi)i=1∞ is equivalent to the canonical ℓq-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)=∥∑i=1nxi∥.
Note that
1≤λ(n)≤n1/p,n∈ℕ,
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=∑i=1nxi,v2=∑i=n+12nxi,…,vj=∑i=(j-1)n+1jnxi,….
Equation (2.3) says that
M-1λ(n)≤∥vj∥≤Mλ(n),j∈ℕ,
and so by the 1-unconditionality of (xi)i=1∞,
1Mλ(n)∥∑j=1mvj∥≤∥∑j=1m∥vj∥-1vj∥≤Mλ(n)∥∑j=1mvj∥,m∈ℕ.
On the other hand, by (2.2) we know that
λ(m)M≤∥∑j=1m∥vj∥-1vj∥≤Mλ(m),m∈ℕ.
If we put these last two inequalities together we obtain
1M2λ(m)λ(n)≤λ(mn)≤M2λ(m)λ(n),m,n∈ℕ.
Substituting in (2.10) integers of the form m=2k and n=2j give
1M2λ(2k)λ(2j)≤λ(2j+k)≤M2λ(2k)λ(2j),k,j∈ℕ.
For k=0,1,2,…, let h(k)=log2λ(2k). From (2.11) it follows that
|h(j)-h(k)-h(j+k)|≤2log2M.
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
for all m,n∈ℕ. Then there is a constant c so that
|sn-cn|≤1,n=1,2,….

Lemma 2.3 yields a constant c so that
|h(k)-ck|≤2log2M,k=1,2,….
In turn, using (2.5) we have
1≤λ(2k)≤2k/p,k=1,2,…
which implies
0≤h(k)≤kp,
and so, combining with (2.15) we obtain that the range of possible values for c is
0≤c≤1p.
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, if 0<c≤1/p there is q∈[p,∞) such that c=1/q. This way we can rewrite (2.15) in the form
|h(k)-kq|≤2log2M,k∈ℕ,
or equivalently,
M-22k/q≤λ(2k)≤2k/qM2,k∈ℕ.
Now, given n∈ℕ we pick the only integer k so that 2k-1≤n≤2k. Then,
λ(2k-1)≤λ(n)≤λ(2k),
and so
M-22-1/qn1/q≤λ(n)≤M221/qn1/q.
If A is any finite subset of ℕ, by (2.3) we have
M-1λ(|A|)≤∥∑j∈Axj∥≤Mλ(|A|),
hence
C-1|A|1/q≤∥∑j∈Axj∥≤C|A|1/q,
where C=M321/q.

To prove the equivalence of (xi)i=1∞ with the canonical basis of ℓq, given any n∈ℕ we let (ai)i=1n be nonnegative scalars such that aiq∈ℚ and ∑i=1naiq=1. Each aiq can be written in the form aiq=mi/m where mi∈ℕ, m is de common denominator of the aiq's, and ∑i=1nmi=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 sets A1,…,An are disjoint and have cardinality |Ai|=mi for each i=1,…,n. Consider the normalized constant coefficient block basic sequence defined as
ui=ci-1∑j∈Aixj,1≤i≤n,
where ci=∥∑j∈Aixk∥. Equation (2.24) yields
C-1mi1/q≤ci≤Cmi1/q,1≤i≤n.
Therefore,
C-1m1/q∥∑i=1n∑j∈Aixj∥≤∥∑i=1naiui∥≤Cm1/q∥∑i=1n∑j∈Aixk∥,
that is,
C-1λ(m)m1/q≤∥∑i=1naiui∥≤Cλ(m)m1/q.
Thus,
1C2M≤∥∑i=1naiui∥≤C2M.
Using (2.2) again, we have
1C2M2≤∥∑i=1naixi∥≤C2M2.
We note that a simple density argument shows that (2.30) holds whenever ∑i=1n|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 ℓq-basis for some p≤q<∞. Since X is not locally convex, we conclude that p≤q<1.

Acknowledgment

The authors would like to acknowledge support from the Spanish Ministerio de Educación y Ciencia Research Project Espacios Topológicos Ordenados: Resultados Analíticos y Aplicaciones Multidisciplinares, reference number MTM2007-62499.

KaltonN. J.PeckN. T.RogersJ. W.RolewiczS.KaltonN. J.Quasi-Banach spacesAokiT.Locally bounded linear topological spacesRolewiczS.On a certain class of linear metric spacesKaltonN. J.Orlicz sequence spaces without local convexityLindenstraussJ.TzafririL.On Orlicz sequence spacesAlbiacF.LeránozC.Uniqueness of symmetric basis in quasi-Banach spacesAlbiacF.LeránozC.Uniqueness of unconditional basis in Lorentz sequence spacesKaltonN. J.LeránozC.WojtaszczykP.Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spacesAlbiacFKaltonN. J.LindenstraussJ.TzafririL.ZippinM.On perfectly homogeneous bases in Banach spaces