Remarks on Gurariĭ Spaces

Abstract

We present selected known results and some new observations, involving Gurariĭ Spaces. A Banach space is Gurariĭ Spaces if it has certain natural extension property for almost isometric embeddings of finite-dimensional spaces. Deleting the word \almost", we get the notion of a strong Gurariĭ Spaces. There exists a unique (up to isometry) separable Gurariĭ Spaces, however strong Gurariĭ Spaces cannot be separable. The structure of the class of non-separable Gurariĭ Spaces seems to be not very well understood. We discuss some of their properties and state some open questions. In particular, we characterize nonseparable Gurariĭ Spaces in terms of skeletons of separable subspaces, we construct a nonseparable Gurariĭ Spaces with a projectional resolution of the identity and we show that no strong Gurariĭ Spaces can be weakly Lindelӧf determined.