Hahn spaces in Fréchet spaces and applications to real sequence spaces
To investigate the extent to which a sequence space is determined by the sequences of \(0\)'s and \(1\)'s that it contains, \textit{G. Bennett}, the first author and the reviewer in [Stud. Math. 149, No. 1, 75--99 (2002; Zbl 0995.46010); addendum ibid. 171, No. 3, 305--309 (2005; \url{doi:10.4064/sm171--3-7})] introduced the notion of Hahn property. As defined by them, a sequence space \(E\) is said to be a Hahn space if, for each FK-space \(F\), \[ \chi\cap E\subset F\Rightarrow E\subset F, \] where \(\chi\) is the set of all sequences of \(0\)'s and \(1\)'s. In the paper under review, this problem is studied in the more general context of FH-spaces. Let \(H\) be a Fréchet space. A Fréchet space \(E\) is called FH-space if \(E\subset H\) and the inclusion map \(i:E\rightarrow H\) is continuous. Hence, FK-spaces are FH-spaces, where \(H=\omega\), the space of all scalar sequences. A tuple \(( H,\psi) \) with \(\psi\subset H\) is called a Hahn tuple if \(\widehat{\psi}\), the intersection of all FH-spaces containing \(\psi\), is an FH-space. (Note that in the prototypical situation with \(H=\omega\) and \(\psi=\chi\) we have \(\widehat{\chi}=\ell_{\infty}\), the FK-space of all bounded sequences.) A subspace \(E\subset H\) is said to be a Hahn space relative to the Hahn tuple \(( H,\psi)\) if the implication \(\psi\cap E\subset F\Rightarrow E\subset F\) holds for each FH-space \(F\). A Hahn space \(E\) is called big if each subspace \(X\) of \(H\) with \(E\subset X\subset\widehat{\psi}\) is a Hahn space\(.\;\) The authors discuss properties of Hahn tuples and Hahn spaces. It is shown that the famous functional analytical characterization of the Hahn property, which holds in FK-spaces, remains valid for FH-spaces: An FH-space \(E\) is a Hahn space relative to the Hahn tuple \(( H,\psi) \) if and only if \(\langle \psi\cap E\rangle \), the linear span of \(\psi\cap E\), is dense and barrelled in \(E\). For a subspace \(E\) of \(H\), the authors define the lower \(\psi\)-zone (the set of all subspaces \(X\subset H\) with \(\psi\cap E\subset X\subset E\)) and the upper \(\psi\)-zone (the set of all subspaces of \(X\subset H\) with \(E\subset X\subset E+\mathbb{K}\cdot\psi\)) and show that every member in the lower and the upper \(\psi\)-zone of a Hahn space \(E\) is also a Hahn space. If, in addition, \(\widehat{\psi}\) belongs to the upper \(\psi\)-zone of \(E\), then \(E\) is a big Hahn space. For example, the sequence space \(bs\oplus\langle e\rangle \) (here, \(bs=\sum^{-1}( \ell_{\infty}) \) is the space of all bounded series and \(e=( 1,1,\dots) \)) is a big Hahn space and each sequence space \(X\) with \(bs\oplus\langle e\rangle \subset X\subset\ell_{\infty}( =\widehat{\chi}) \) is a Hahn space. This fact is generalized as follows. By making use of results obtained in the general setting of Hahn tuples, the authors prove the following theorem: If a summability matrix \(A=( a_{nk}) \) is \(\chi\)-regular, (i.e., \(A\) is normal, \(a_{nk} \in\mathbb{Z}\), \(( a_{nn}) \in\ell_{\infty}\) and \(\sum _{k=1}^{n-1}| a_{nk}| \leq| a_{nn}| \) \(( n,k\in\mathbb{N}))\), then \(A( \ell_{\infty}) +\langle e\rangle \) is a big Hahn space. The last two sections of the paper are devoted to examples of Hahn tuples. In the case \(H=\omega\), the Hahn tuples \(( \omega,T) \) with \(T\in\{ \chi\cap c_{C_{1}},\chi\cap P,\chi\cap ac\} \) are discussed, where \(c_{C_{1}},P\) and \(ac\) are the space of all Cesàro summable sequences, the space of all periodical sequences and the space of all almost convergent sequences, respectively. For the Banach space \(H:=B[ 0,1] \) of all bounded real functions on \([ 0,1]\), the authors consider the Hahn tuples \(( H,\chi( [ 0,1] ,\mathcal{A}) ) \), \(( H,\mathcal{F}_{r}) \) and \(( H,\mathcal{F}) \), where \(\chi( [ 0,1] ,\mathcal{A}) :=\{ \chi_{A}\mid A\in\mathcal{A}\} \) and \(\mathcal{A}=\mathcal{B}[ 0,1] \), the \(\sigma\)-algebra of all Borel sets of \([ 0,1] \), or \(\mathcal{A}=\mathcal{P}[ 0,1] \), the power set of \([ 0,1] \), \(\mathcal{F} _{r}:=\{ f\cdot\chi_{A}\mid f\in P_{r}[ 0,1] ,\text{ } A\in\mathcal{B}[ 0,1] \} \), \(\mathcal{F}:=\bigcup _{r\in\mathbb{N}}\mathcal{F}_{r}\), and \(P_{r}[ 0,1] \) is the set of all real polynomials on \([ 0,1] \) of degree \(r\) or less.
scientific article; zbMATH DE number 6341730
scientific article