Embedding theorems in functional analysis (Q2712745)

This is a survey article describing a variety of embedding theorems and their consequences. For example, the Banach-Mazur theorem that every separable Banach space is a closed subspace of \(C[0,1]\) has a consequence that every separable Banach space may be renormed in a strictly convex manner. Variations of this theme are shown to throw light on some open problems in the structure theory of Banach and nuclear Fréchet spaces. For example, Pełczynski asked whether complemented subspaces of nuclear Fréchet spaces with basis necessarily have a basis. It is now known that this is not the case for the larger class of Fréchet-Schwartz spaces. A number of other questions are posed, such asNEWLINENEWLINENEWLINE``Can every Fréchet-Schwartz space be embedded in another such space with an unconditional bases?''NEWLINENEWLINENEWLINEFor this one, but not for all, an affirmative answer is provided.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].