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The topological product structure of systems of Lebesgue spaces. (English) Zbl 0734.46013
We consider the Banach spaces $$\ell^ p$$ and for $$q<p$$ the subspace $$\ell^ p_ q$$ of $$\ell^ p$$ consisting of q-summable sequences. According to the Anderson-Kadec theorem every $$\ell^ p$$ is homeomorphic to the countable product of lines $${\mathbb{R}}^ N$$. If A is a countable infinite set then we define the following subspaces of the topological Hilbert space $${\mathbb{R}}^ A:\Sigma$$ (A) consists of all bounded $$x=(x_ a)_{a\in A}$$ in $${\mathbb{R}}^ A$$ and $$\sigma$$ (A) consists of all $$x\in {\mathbb{R}}^ A$$ such that $$x_ a=0$$ for all but finitely many $$a\in A$$. It is easily seen that every $$\ell^ p_ q$$ is a so-called $$\sigma$$ Z-set in $$\ell^ p$$ and one may expect that $$\ell^ p_ q$$ is a Z-absorber, i.e. the pair $$(\ell^ p,\ell^ p_ q)$$ is homeomorphic to the pairs $$({\mathbb{R}}^ N\times {\mathbb{R}}^ N,{\mathbb{R}}^ N\times \Sigma (N))$$ and $$({\mathbb{R}}^ N\times {\mathbb{R}}^ N,{\mathbb{R}}^ N\times \sigma (N))$$. This leads to the following defnitions. Let $$p\in (0,\infty]$$ and let A be a countable dense subset of the interval (0,p). If $$q\in (0,p)$$ then we have $Z^ p_ q={\mathbb{R}}^{(0,q]\cap A}\times \Sigma ((q,p)\cap A)\subset {\mathbb{R}}^ A\text{ and } \zeta^ p_ q={\mathbb{R}}^{(0,q]\cap A}\times \sigma ((q,p)\cap A)\subset {\mathbb{R}}^ A.$ The main result is that if B is an arbitrary countable dense subset of (0,p) then there exist homeomorphisms $$\alpha$$ and $$\beta$$ from $${\mathbb{R}}^ A$$ onto $$\ell^ p$$ such that for every $$q\in B$$, $\alpha (Z^ p_ q)=\ell^ p_ q\text{ and } \beta (\zeta^ p_ q)=\ell^ p_ q.$ This means that the connection is established between the structure that the system of subspaces $$\ell^ p_ q$$ puts on the space $$\ell^ p$$ and a structure that finds its origin in the underlying topological product.
Similar results are obtained for the function spaces $$L^ p$$.
Reviewer: J.J.Dijkstra

##### MSC:
 46B45 Banach sequence spaces 46A45 Sequence spaces (including Köthe sequence spaces) 46B25 Classical Banach spaces in the general theory
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##### References:
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