Let $X$ be a separable Banach space. Is this property equivalent to the approximation property?


There exists a chain $X\_n$ of finite-dimensional subspaces of $X$, each being a range of some projection $P\_n$, such that $\bigcup X\_n$ is dense in $X$ and for every $x\in X$ we have $P\_nx \to x$ as $n\to \infty$?


Does this depend on the choice of a chain of finite dimensional subspaces with dense union?



Approximation property of a Banach space in terms of finite-rank projections
It seems to me that the property which you describe is equivalent to the $\pi$-property. An interesting example, showing that it does not follow from the approximation property was discovered by Charles Read (1958-2015). Unfortunately Charles never published this nice paper, but you can find it on my web page: <http://facpub.stjohns.edu/ostrovsm/ReadDifferentFormsOfAP1989UnpublishedManuscript.pdf>


The second question was already answered by Mikael de la Salle (see above).


