Weak and quasi approximation properties in Banach spaces
The authors say that a Banach space \(X\) has the [bounded] weak approximation property if every compact operator \(T: X\to X\) is the limit of a [bounded] net of finite-rank operators for the topology of uniform convergence on compact sets, and they say that \(X\) has the quasi approximation property if the convergence takes place with respect to the operator norm. The paper presents some observations concerning these new notions that are elementary, well known or follow the same proofs as their classical counterparts (e.g., the weak approximation property of \(X^*\) implies that of \(X\)). No result is nontrivially new, and what the authors call their own proof of Lemma~3.1, a result due to Grothendieck, is actually a copy of the proof of Proposition~1.e.3 in \textit{J.~Lindenstrauss}' and \textit{L.~Tzafriri}'s classical book [Classical Banach Spaces~I. Springer-Verlag (1977; Zbl 0362.46013); re-edited 1996 (Zbl 0852.46015)]. For more serious recent attempts to weaken the approximation property see [\textit{E.~Odell, H.-O.~Tylli}, Trans.\ Am.\ Math.\ Soc.\ 357, No. 3, 1125--1159 (2005; Zbl 1067.46028)] or [\textit{Å.~Lima, E.~Oja}, Math.\ Ann.\ 333, No. 3, 471--484 (2005; Zbl 1097.46012)].
scientific article; zbMATH DE number 5016213
scientific article