The topological product structure of systems of Lebesgue spaces (Q810821)

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 \(L^ p\).