For Banach spaces, the existence of a Schauder basis implies that this space has the approximation property.
Since both the notion of Schauder bases and of the approximation property are well established for locally convex spaces, it seems natural to ask how far the above result can be generalised. It is not so difficult to show that every Frechet space with a Schauder basis has the approximation property.
My main question is: What is the general situation?
It seems that the above should also be true for barrelled locally convex spaces. Has this case been considered in the literature?
Is there a locally convex space with a Schauder basis but without the approximation property?
