Topologists are interested in the classification of topological spaces. Classifying topological spaces up to homeomorphism or homotopy type etc., is the ultimate goal for a topologist. We are interested here in classifying spaces up to homeomorphism or linear-homeomorphism type of their function spaces . In its full generality this program would be much too complicated and the complete picture is presently beyond our reach. But for function spaces of low Borel complexity some definitive results are known, and it is our aim to discuss them here.
We say that spaces and are -equivalent provided that and are linearly homeomorphic. Notation: .
Homeomorphic spaces are obviously -equivalent. But the converse need not be true. Let and . Then evidently, and are not homeomorphic. However, they are -equivalent. Indeed, define by
We say that and are -equivalent provided that are are homeomorphic as topological spaces. Notation: .
Even for simple spaces it is in general difficult to decide whether they are - or -equivalent. By Bessaga and Peczynski  there are countable compact spaces and for which the Banach spaces and are not linearly homeomorphic. An application of the Closed Graph Theorem shows that if and are linearly homeomorphic then so are and (the same linear map does the job in both cases). So the examples of Bessaga and Peczynski are not -equivalent. This suggests the question whether they are -equivalent. We will come back to this below.
Arhangelski  proved that if is compact and is linearly homeomorphic to then is compact. As a consequence, and are not linearly homeomorphic. But they are homeomorphic, as was shown by Gul'ko and Khmyleva .
Results in the same spirit were obtained by various authors. Pestov  proved that if and are linearly homeomorphic then and have the same dimension. So and are not linearly homeomorphic. Observe that by the famous result of Miljutin , all Banach spaces with uncountable and compact metrizable are linearly homeomorphic. Hence and are linearly homeomorphic, but and are not. For another result in the same spirit, see Baars, de Groot and Pelant .