Seminar about the Bounded Approximation Property in Fréchet Spaces

The purpose of this seminar, which was presented at the Universitat Politècnica de València in late 2012, is to explain several results concerning the bounded approximation property for Fréchet spaces. We give a full detailed proof of an important result due to Pełczyński that asserts that every separable Fréchet space with the bounded approximation property is isomorphic to a complemented subspace of a Fréchet space with a Schauder basis. We also explain Vogt's example of a nuclear Fréchet 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 Fréchet function space without the bounded approximation property. The relation of the bounded approximation property for Fréchet spaces with a continuous norm and the countably normable spaces, including several results due to Dubinski and Vogt, is also explained.

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