On Perfectly Homogeneous Bases in Quasi-Banach Spaces
For the unit vector basis of 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 -basis or the canonical -basis for some . In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of for 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 . A (real) quasi-normed space is a locally bounded topological vector space. This is equivalent to saying that the topology on is induced by a quasi-norm , that is, a map satisfying(i) if and only if ;(ii) if ;(iii) there is a constant so that for any and we have
The best constant in inequality (1.1) is called the modulus of concavity of the quasi-norm. If , the quasi-norm is a norm. A quasi-norm on is -subadditive if A theorem by Aoki  and Rolewicz  asserts that every quasi-norm has an equivalent -subadditive quasi-norm, where is given by . A -subadditive quasi-norm induces an invariant metric on by the formula . The space is called quasi-Banach space if 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 of a quasi-Banach space is symmetric if is equivalent to for any permutation of . Symmetric bases are unconditional and so there exists a nonnegative constant such that for all the inequality holds for any bounded sequence . The least such constant is called the unconditional constant of .
For instance, the canonical basis of the spaces for is symmetric and -unconditional. What is more, it is the only symmetric basis of up to equivalence, that is, whenever is another normalized symmetric basis of , there is a constant such that for any finitely nonzero sequence of scalars [6, 7].
The spaces for 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 , as was recently proved in . 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 for 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 be a basis for a quasi-Banach space . A block basic sequence of , is said to be a constant coefficient block basic sequence if for each there is a constant so that or for .
Definition 2.1. A basis of a quasi-Banach space is almost perfectly homogeneous if every normalized constant coefficient block basic sequence of is equivalent to .
Let us notice that using a uniform boundedness argument we obtain that, in fact, if is almost perfectly homogeneous then it is uniformly equivalent to all its normalized constant coefficient block basic sequences. That is, there is a constant such that for any normalized constant coefficient block basic sequence of we have for all choices of scalars and . Equation (2.2) also yields that for any increasing sequence of integers ,
This is our main result (cf. ).
Theorem 2.2. Let be a nonlocally convex quasi-Banach space with normalized basis . Suppose that is almost perfectly homogeneous. Then is equivalent to the canonical basis of for some .
Proof. Let be the modulus of concavity of the quasi-norm. Since is nonlocally convex, . By the Aoki-Rolewicz theorem we can assume that the quasi-norm is -subadditive for such that . We will show that is equivalent to the canonical -basis for some .
By renorming, without loss of generality we can assume to be -unconditional. For each put, Note that and that, by the -unconditionality of the basis, the sequence is nondecreasing.
We are going to construct disjoint blocks of length of the basis as follows: Equation (2.3) says that and so by the -unconditionality of , On the other hand, by (2.2) we know that If we put these last two inequalities together we obtain Substituting in (2.10) integers of the form and give For let . From (2.11) it follows that We need the following well-known lemma from real analysis.
Lemma 2.3. Suppose that is a sequence of real numbers such that for all . Then there is a constant so that
Lemma 2.3 yields a constant so that In turn, using (2.5) we have which implies and so, combining with (2.15) we obtain that the range of possible values for is If then would be (uniformly) bounded and so would be equivalent to the canonical basis of , a contradiction with the local nonconvexity of . Otherwise, if there is such that . This way we can rewrite (2.15) in the form or equivalently, Now, given we pick the only integer so that . Then, and so If is any finite subset of , by (2.3) we have hence where .
To prove the equivalence of with the canonical basis of , given any we let be nonnegative scalars such that and . Each can be written in the form where , is de common denominator of the 's, and .
Let be interval of natural numbers and for let be the interval of natural numbers . The sets are disjoint and have cardinality for each . Consider the normalized constant coefficient block basic sequence defined as where Equation (2.24) yields Therefore, that is, Thus, Using (2.2) again, we have We note that a simple density argument shows that (2.30) holds whenever (i.e., without the assumption that is rational), and this completes the proof that is equivalent to the canonical -basis for some . Since is not locally convex, we conclude that .
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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