  The purpose of this seminar, which was presented at the Universitat
Polit\`ecnica de Val\`encia in late 2012, is to explain several results
concerning the bounded approximation property for Fr\'echet spaces. We give a
full detailed proof of an important result due to Pe\l czy\'nski that asserts
that every separable Fr\'echet space with the bounded approximation property is
isomorphic to a complemented subspace of a Fr\'echet space with a Schauder
basis. We also explain Vogt's example of a nuclear Fr\'echet space without the
bounded approximation property. This example is simpler than the original
counterexample due to Dubinski. These examples solved a long standing problem
of Grothendieck. Vogt obtained later another simple example of a nuclear
Fr\'echet function space without the bounded approximation property. The
relation of the bounded approximation property for Fr\'echet spaces with a
continuous norm and the countably normable spaces, including several results
due to Dubinski and Vogt, is also explained.
