On universal Banach spaces of density continuum
On universal Banach spaces of density continuum
复制标题
密度连续统的通用巴纳赫空间
DOI：
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发表时间：
2010
期刊：
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通讯作者：
P. Koszmider
中科院分区：
文献类型：
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作者：
C. Brech;P. Koszmider
我们考虑是否存在密度连续的Banach空间X使得每个至多连续的密度的Banach空间同构嵌入到X中(称为密度c的泛Banach空间)。众所周知，如果我们假设连续统假设，ℓ∞/c0就是这样一个空间。在本文的主要结果中，我们证明了不存在密度为c的泛Banach空间，这与集合论中通常的公理是一致的。因此，密度为c的泛Banach空间的存在是不能用集合论的通常公理来判定的。我们还证明了密度为c的泛Banach空间的存在是一致的，但ℓ∞/c0不在其中。这依赖于证明C([0，c])到ℓ∞/c0的同构嵌入不存在的一致性。
We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density at most continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that ℓ∞/c0 is such a space if we assume the continuum hypothesis. Some additional set-theoretic assumption is indeed needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory.We also prove that it is consistent that there are universal Banach spaces of density c, but ℓ∞/c0 is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0, c]) into ℓ∞/c0.