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The purpose of this seminar, which was presented at the UniversitatPolit\`ecnica de Val\`encia in late 2012, is to explain several resultsconcerning the bounded approximation property for Fr\'echet spaces. We give afull detailed proof of an important result due to Pe\l czy\'nski that assertsthat every separable Fr\'echet space with the bounded approximation property isisomorphic to a complemented subspace of a Fr\'echet space with a Schauderbasis. We also explain Vogt's example of a nuclear Fr\'echet space without thebounded approximation property. This example is simpler than the originalcounterexample due to Dubinski. These examples solved a long standing problemof Grothendieck. Vogt obtained later another simple example of a nuclearFr\'echet function space without the bounded approximation property. Therelation of the bounded approximation property for Fr\'echet spaces with acontinuous norm and the countably normable spaces, including several resultsdue to Dubinski and Vogt, is also explained.

Seminar about the Bounded Approximation Property in Fr\'echet Spaces