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An embedding property of sequence spaces related to Meyer-König and Zeller type theorems. (English) Zbl 0632.46007

For BK spaces Y, Z, write \(Y<Z\), if \(Y\subset Z\) and if \(X=Z\) for every FK space X with \(\phi\subset X\) and \(X+Y\subset Z\). If U is a BK space contained in a sequence space V, then the embedding \(U\to V\) is said to have MKWZ (Meyer-König-Wilansky-Zeller property) if \(X\cap V\subset U\) implies that \(X\cap U\) is closed in X for each FK space X.
Theorem 1 shows that MKWZ is implied by a certain gliding humps property of \(U\to V\), if \(\phi\subset U\) and dim(U/\({\bar \phi}\))\(<\infty\). Several known theorems of Meyer-König and Zeller type follow as corollaries.
Let now Y, Z be BK spaces with \(\phi\subset Y\subset Z\); suppose that the unit vectors form a basis of Y, Z and that the closure of \(\phi\) in the \(\gamma\)-dual \(Z^{\gamma}\) of Z has finite codimension in \(Z^{\gamma}\). The main theorem 2 then states that \(Y<Z\) if and only if \(Z^{\gamma}\to Y^{\gamma}\) has MKWZ. The results are applied to the case that Y is weakly compactly embedded in Z and to classical sequence spaces.
Reviewer: W.Roelcke

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46B25 Classical Banach spaces in the general theory
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