Bases and Isomorphisms of Whitney Spaces

We consider results related to bases and isomorphisms of Whitney spaces E(K) and the extension property of compact sets K. The first two sections discuss the notion of a topological basis and its importance in analysis. Then we consider model spaces of infinitely differentiable functions that may occur in applications. Our main interest is in Whitney spaces E(K) and the extension property of compact sets, that is, the existence of a continuous linear extension operator from E(K) to the space of infinitely differentiable functions on the whole Euclidean space. In Section 5 we consider what we believe to be the main methods for constructing such an operator. Section 6 contains some geometric and other conditions characterizing the extension property. Sections 7-11 are devoted to bases in spaces of infinitely differentiable functions, Whitney spaces, and restriction spaces, with an emphasis on the author's results obtained using the method of local interpolations. The final sections present results related to the isomorphic classification of these spaces. We consider the counting linear topological invariants (mainly the diametrical dimension), interpolation invariants (mainly generalizations of the dominated norm property), and compound invariants, which reduce to the computation of diametrical dimension for so-called synthetic neighborhoods. Various families of the continuum cardinality of pairwise non-isomorphic spaces are presented. Finally, some open problems are proposed. The review contains many examples, both classic and new.