Abstract and Applied Analysis
Volume 2009 (2009), Article ID 865371, 7 pages
doi:10.1155/2009/865371
Research Article

On Perfectly Homogeneous Bases in Quasi-Banach Spaces

F. Albiac and C. Leránoz

Departamento de Matemáticas, Universidad Pública de Navarra, 31006 Pamplona, Spain

Received 22 April 2009; Accepted 3 June 2009

Academic Editor: Simeon Reich

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

Abstract

For 0 < 𝑝 < ∞ the unit vector basis of ℓ 𝑝 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 𝑐 0 -basis or the canonical ℓ 𝑝 -basis for some 1 ≤ 𝑝 < ∞ . In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of ℓ 𝑝 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

(i) ‖ 𝑥 ‖ = 0 if and only if 𝑥 = 0 ;(ii) ‖ 𝛼 𝑥 ‖ = | 𝛼 | ‖ 𝑥 ‖ if 𝛼 ∈ ℝ , 𝑥 ∈ 𝑋 ;(iii) there is a constant 𝜅 ≥ 1 so that for any 𝑥 1 and 𝑥 2 ∈ 𝑋 we have ‖ 𝑥 1 + 𝑥 2 ‖ ≤ 𝜅 ‖ 𝑥 1 ‖ + ‖ 𝑥 2 ‖ . ( 1 . 1 )

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 ‖ 𝑥 1 + 𝑥 2 ‖ 𝑝 ≤ ‖ 𝑥 1 ‖ 𝑝 + ‖ 𝑥 2 ‖ 𝑝 , 𝑥 1 , 𝑥 2 ∈ 𝑋 . ( 1 . 2 ) 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 = 2 1 / 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 ( 𝑥 𝑛 ) ∞ 𝑛 = 1 of a quasi-Banach space 𝑋 is symmetric if ( 𝑥 𝑛 ) ∞ 𝑛 = 1 is equivalent to ( 𝑥 𝜋 ( 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 𝑎 𝑛 𝑥 𝑛 the inequality ‖ ‖ ‖ ‖ ∞ 𝑛 = 1 𝜃 𝑛 𝑎 𝑛 𝑥 𝑛 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ≤ 𝐾 ∞ 𝑛 = 1 𝑎 𝑛 𝑥 𝑛 ‖ ‖ ‖ ‖ ( 1 . 3 ) holds for any bounded sequence ( 𝜃 𝑛 ) ∞ 𝑛 = 1 ∈ 𝐵 ℓ ∞ . The least such constant 𝐾 is called the unconditional constant of ( 𝑥 𝑛 ) ∞ 𝑛 = 1 .

For instance, the canonical basis of the spaces ℓ 𝑝 for 0 < 𝑝 < ∞ is symmetric and 1 -unconditional. What is more, it is the only symmetric basis of ℓ 𝑝 up to equivalence, that is, whenever ( 𝑥 𝑛 ) ∞ 𝑛 = 1 is another normalized symmetric basis of ℓ 𝑝 , there is a constant 𝐶 such that 1 𝐶 ∞ 𝑛 = 1 | 𝑎 𝑛 | 𝑝 1 / 𝑝 ≤ ‖ ‖ ‖ ‖ ∞ 𝑛 = 1 𝑎 𝑛 𝑥 𝑛 ‖ ‖ ‖ ‖ ≤ 𝐶 ∞ 𝑛 = 1 | 𝑎 𝑛 | 𝑝 1 / 𝑝 , ( 1 . 4 ) for any finitely nonzero sequence of scalars ( 𝑎 𝑛 ) ∞ 𝑛 = 1 [6, 7].

The spaces ℓ 𝑝 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 ℓ 1 , 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 ℓ 𝑝 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 ( 𝑥 𝑖 ) ∞ 𝑖 = 1 be a basis for a quasi-Banach space 𝑋 . A block basic sequence ( 𝑢 𝑛 ) ∞ 𝑛 = 1 of ( 𝑥 𝑖 ) ∞ 𝑖 = 1 , 𝑢 𝑛 = 𝑝 𝑛 𝑝 𝑛 − 1 + 1 𝑎 𝑖 𝑥 𝑖 , ( 2 . 1 ) is said to be a constant coefficient block basic sequence if for each 𝑛 there is a constant 𝑐 𝑛 so that 𝑎 𝑖 = 𝑐 𝑛 or 𝑎 𝑖 = 0 for 𝑝 𝑛 − 1 + 1 ≤ 𝑖 ≤ 𝑝 𝑛 .

Definition 2.1. A basis ( 𝑥 𝑖 ) ∞ 𝑖 = 1 of a quasi-Banach space 𝑋 is almost perfectly homogeneous if every normalized constant coefficient block basic sequence of ( 𝑥 𝑖 ) ∞ 𝑖 = 1 is equivalent to ( 𝑥 𝑖 ) ∞ 𝑖 = 1 .

Let us notice that using a uniform boundedness argument we obtain that, in fact, if ( 𝑥 𝑖 ) ∞ 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 ( 𝑢 𝑛 ) ∞ 𝑛 = 1 of ( 𝑥 𝑖 ) ∞ 𝑖 = 1 we have 𝑀 − 1 ‖ ‖ ‖ ‖ 𝑛 𝑘 = 1 𝑎 𝑘 𝑥 𝑘 ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ 𝑛 𝑘 = 1 𝑎 𝑘 𝑢 𝑘 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ≤ 𝑀 𝑛 𝑘 = 1 𝑎 𝑘 𝑥 𝑘 ‖ ‖ ‖ ‖ , ( 2 . 2 ) for all choices of scalars ( 𝑎 𝑘 ) 𝑛 𝑘 = 1 and 𝑛 ∈ ℕ . Equation (2.2) also yields that for any increasing sequence of integers ( 𝑘 𝑗 ) ∞ 𝑗 = 1 , 𝑀 − 1 ‖ ‖ ‖ ‖ 𝑛 𝑗 = 1 𝑥 𝑗 ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ 𝑛 𝑗 = 1 𝑥 𝑘 𝑗 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ≤ 𝑀 𝑛 𝑗 = 1 𝑥 𝑗 ‖ ‖ ‖ ‖ . ( 2 . 3 )

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

Theorem 2.2. Let 𝑋 be a nonlocally convex quasi-Banach space with normalized basis ( 𝑥 𝑖 ) ∞ 𝑖 = 1 . Suppose that ( 𝑥 𝑖 ) ∞ 𝑖 = 1 is almost perfectly homogeneous. Then ( 𝑥 𝑖 ) ∞ 𝑖 = 1 is equivalent to the canonical basis of ℓ 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 for 0 < 𝑝 < 1 such that 𝜅 = 2 1 / 𝑝 − 1 . We will show that ( 𝑥 𝑖 ) ∞ 𝑖 = 1 is equivalent to the canonical ℓ 𝑞 -basis for some 𝑝 ≤ 𝑞 < 1 .
By renorming, without loss of generality we can assume ( 𝑥 𝑖 ) ∞ 𝑖 = 1 to be 1 -unconditional. For each 𝑛 put, ‖ ‖ ‖ ‖ 𝜆 ( 𝑛 ) = 𝑛 𝑖 = 1 𝑥 𝑖 ‖ ‖ ‖ ‖ . ( 2 . 4 ) Note that 1 ≤ 𝜆 ( 𝑛 ) ≤ 𝑛 1 / 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 ( 𝑥 𝑖 ) ∞ 𝑖 = 1 as follows: 𝑣 1 = 𝑛 𝑖 = 1 𝑥 𝑖 , 𝑣 2 = 2 𝑛 𝑖 = 𝑛 + 1 𝑥 𝑖 , … , 𝑣 𝑗 = 𝑗 𝑛 𝑖 = ( 𝑗 − 1 ) 𝑛 + 1 𝑥 𝑖 , … . ( 2 . 6 ) Equation (2.3) says that 𝑀 − 1 𝜆 ( 𝑛 ) ≤ ‖ 𝑣 𝑗 ‖ ≤ 𝑀 𝜆 ( 𝑛 ) , 𝑗 ∈ ℕ , ( 2 . 7 ) and so by the 1 -unconditionality of ( 𝑥 𝑖 ) ∞ 𝑖 = 1 , 1 ‖ ‖ ‖ ‖ 𝑀 𝜆 ( 𝑛 ) 𝑚 𝑗 = 1 𝑣 𝑗 ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ 𝑚 𝑗 = 1 ‖ 𝑣 𝑗 ‖ − 1 𝑣 𝑗 ‖ ‖ ‖ ‖ ≤ 𝑀 ‖ ‖ ‖ ‖ 𝜆 ( 𝑛 ) 𝑚 𝑗 = 1 𝑣 𝑗 ‖ ‖ ‖ ‖ , 𝑚 ∈ ℕ . ( 2 . 8 ) On the other hand, by (2.2) we know that 𝜆 ( 𝑚 ) 𝑀 ≤ ‖ ‖ ‖ ‖ 𝑚 𝑗 = 1 ‖ 𝑣 𝑗 ‖ − 1 𝑣 𝑗 ‖ ‖ ‖ ‖ ≤ 𝑀 𝜆 ( 𝑚 ) , 𝑚 ∈ ℕ . ( 2 . 9 ) If we put these last two inequalities together we obtain 1 𝑀 2 𝜆 ( 𝑚 ) 𝜆 ( 𝑛 ) ≤ 𝜆 ( 𝑚 𝑛 ) ≤ 𝑀 2 𝜆 ( 𝑚 ) 𝜆 ( 𝑛 ) , 𝑚 , 𝑛 ∈ ℕ . ( 2 . 1 0 ) Substituting in (2.10) integers of the form 𝑚 = 2 𝑘 and 𝑛 = 2 𝑗 give 1 𝑀 2 𝜆 2 𝑘 𝜆 2 𝑗 2 ≤ 𝜆 𝑗 + 𝑘 ≤ 𝑀 2 𝜆 2 𝑘 𝜆 2 𝑗 , 𝑘 , 𝑗 ∈ ℕ . ( 2 . 1 1 ) For 𝑘 = 0 , 1 , 2 , … , let ℎ ( 𝑘 ) = l o g 2 𝜆 ( 2 𝑘 ) . From (2.11) it follows that | | | | ℎ ( 𝑗 ) − ℎ ( 𝑘 ) − ℎ ( 𝑗 + 𝑘 ) ≤ 2 l o g 2 𝑀 . ( 2 . 1 2 ) We need the following well-known lemma from real analysis.
Lemma 2.3. Suppose that ( 𝑠 𝑛 ) ∞ 𝑛 = 1 is a sequence of real numbers such that | | 𝑠 𝑚 + 𝑛 − 𝑠 𝑚 − 𝑠 𝑛 | | ≤ 1 ( 2 . 1 3 ) for all 𝑚 , 𝑛 ∈ ℕ . Then there is a constant 𝑐 so that | | 𝑠 𝑛 | | − 𝑐 𝑛 ≤ 1 , 𝑛 = 1 , 2 , … . ( 2 . 1 4 )
Lemma 2.3 yields a constant 𝑐 so that | | | | ℎ ( 𝑘 ) − 𝑐 𝑘 ≤ 2 l o g 2 𝑀 , 𝑘 = 1 , 2 , … . ( 2 . 1 5 ) In turn, using (2.5) we have 2 1 ≤ 𝜆 𝑘 ≤ 2 𝑘 / 𝑝 , 𝑘 = 1 , 2 , … ( 2 . 1 6 ) which implies 𝑘 0 ≤ ℎ ( 𝑘 ) ≤ 𝑝 , ( 2 . 1 7 ) and so, combining with (2.15) we obtain that the range of possible values for 𝑐 is 1 0 ≤ 𝑐 ≤ 𝑝 . ( 2 . 1 8 ) If 𝑐 = 0 then ( 𝜆 ( 𝑛 ) ) ∞ 𝑛 = 1 would be (uniformly) bounded and so ( 𝑥 𝑖 ) ∞ 𝑖 = 1 would be equivalent to the canonical basis of 𝑐 0 , 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 ( 𝑘 ) − 𝑞 | | | | ≤ 2 l o g 2 𝑀 , 𝑘 ∈ ℕ , ( 2 . 1 9 ) or equivalently, 𝑀 − 2 2 𝑘 / 𝑞 2 ≤ 𝜆 𝑘 ≤ 2 𝑘 / 𝑞 𝑀 2 , 𝑘 ∈ ℕ . ( 2 . 2 0 ) Now, given 𝑛 ∈ ℕ we pick the only integer 𝑘 so that 2 𝑘 − 1 ≤ 𝑛 ≤ 2 𝑘 . Then, 𝜆 2 𝑘 − 1 2 ≤ 𝜆 ( 𝑛 ) ≤ 𝜆 𝑘 , ( 2 . 2 1 ) and so 𝑀 − 2 2 − 1 / 𝑞 𝑛 1 / 𝑞 ≤ 𝜆 ( 𝑛 ) ≤ 𝑀 2 2 1 / 𝑞 𝑛 1 / 𝑞 . ( 2 . 2 2 ) If 𝐴 is any finite subset of ℕ , by (2.3) we have 𝑀 − 1 𝜆 | | 𝐴 | | ≤ ‖ ‖ ‖ ‖ 𝑗 ∈ 𝐴 𝑥 𝑗 ‖ ‖ ‖ ‖ | | 𝐴 | | ≤ 𝑀 𝜆 , ( 2 . 2 3 ) hence 𝐶 − 1 | 𝐴 | 1 / 𝑞 ≤ ‖ ‖ ‖ ‖ 𝑗 ∈ 𝐴 𝑥 𝑗 ‖ ‖ ‖ ‖ ≤ 𝐶 | 𝐴 | 1 / 𝑞 , ( 2 . 2 4 ) where 𝐶 = 𝑀 3 2 1 / 𝑞 .
To prove the equivalence of ( 𝑥 𝑖 ) ∞ 𝑖 = 1 with the canonical basis of ℓ 𝑞 , given any 𝑛 ∈ ℕ we let ( 𝑎 𝑖 ) 𝑛 𝑖 = 1 be nonnegative scalars such that 𝑎 𝑞 𝑖 ∈ ℚ and ∑ 𝑛 𝑖 = 1 𝑎 𝑞 𝑖 = 1 . Each 𝑎 𝑞 𝑖 can be written in the form 𝑎 𝑞 𝑖 = 𝑚 𝑖 / 𝑚 where 𝑚 𝑖 ∈ ℕ , 𝑚 is de common denominator of the 𝑎 𝑞 𝑖 's, and ∑ 𝑛 𝑖 = 1 𝑚 𝑖 = 𝑚 .
Let 𝐴 1 be interval of natural numbers [ 1 , 𝑚 1 ] and for 𝑗 = 2 , … , 𝑛 let 𝐴 𝑖 be the interval of natural numbers [ 𝑚 1 + ⋯ + 𝑚 𝑖 − 1 + 1 , 𝑚 1 + ⋯ + 𝑚 𝑖 ] . The sets 𝐴 1 , … , 𝐴 𝑛 are disjoint and have cardinality | 𝐴 𝑖 | = 𝑚 𝑖 for each 𝑖 = 1 , … , 𝑛 . Consider the normalized constant coefficient block basic sequence defined as 𝑢 𝑖 = 𝑐 𝑖 − 1 𝑗 ∈ 𝐴 𝑖 𝑥 𝑗 , 1 ≤ 𝑖 ≤ 𝑛 , ( 2 . 2 5 ) where 𝑐 𝑖 ∑ = ‖ 𝑗 ∈ 𝐴 𝑖 𝑥 𝑘 ‖ . Equation (2.24) yields 𝐶 − 1 𝑚 𝑖 1 / 𝑞 ≤ 𝑐 𝑖 ≤ 𝐶 𝑚 𝑖 1 / 𝑞 , 1 ≤ 𝑖 ≤ 𝑛 . ( 2 . 2 6 ) Therefore, 𝐶 − 1 𝑚 1 / 𝑞 ‖ ‖ ‖ ‖ 𝑛 𝑖 = 1 𝑗 ∈ 𝐴 𝑖 𝑥 𝑗 ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ 𝑛 𝑖 = 1 𝑎 𝑖 𝑢 𝑖 ‖ ‖ ‖ ‖ ≤ 𝐶 𝑚 1 / 𝑞 ‖ ‖ ‖ ‖ 𝑛 𝑖 = 1 𝑗 ∈ 𝐴 𝑖 𝑥 𝑘 ‖ ‖ ‖ ‖ , ( 2 . 2 7 ) that is, 𝐶 − 1 𝜆 ( 𝑚 ) 𝑚 1 / 𝑞 ≤ ‖ ‖ ‖ ‖ 𝑛 𝑖 = 1 𝑎 𝑖 𝑢 𝑖 ‖ ‖ ‖ ‖ 𝜆 ≤ 𝐶 ( 𝑚 ) 𝑚 1 / 𝑞 . ( 2 . 2 8 ) Thus, 1 𝐶 2 𝑀 ≤ ‖ ‖ ‖ ‖ 𝑛 𝑖 = 1 𝑎 𝑖 𝑢 𝑖 ‖ ‖ ‖ ‖ ≤ 𝐶 2 𝑀 . ( 2 . 2 9 ) Using (2.2) again, we have 1 𝐶 2 𝑀 2 ≤ ‖ ‖ ‖ ‖ 𝑛 𝑖 = 1 𝑎 𝑖 𝑥 𝑖 ‖ ‖ ‖ ‖ ≤ 𝐶 2 𝑀 2 . ( 2 . 3 0 ) We note that a simple density argument shows that (2.30) holds whenever ∑ 𝑛 𝑖 = 1 | 𝑎 𝑖 | 𝑞 = 1 (i.e., without the assumption that | 𝑎 𝑖 | 𝑞 is rational), and this completes the proof that ( 𝑥 𝑖 ) ∞ 𝑖 = 1 is equivalent to the canonical ℓ 𝑞 -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.

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